Arsenal continue to set the pace at the top of the Premier League after beating Hull City, while nearest pursuers Chelsea and Manchester City recorded contrasting wins.
Rene Meulensteen's first match in charge of Fulham ended in a derby defeat by Tottenham Hotspur as Newcastle United came unstuck at Swansea City.
Aston Villa got the better of a five-goal affair at Southampton, but there was a scoreless stalemate between Stoke City and Cardiff City.
Bryan Oviedo scored an 86th-minute goal as Everton won at Old Trafford for the first time since 1992 to ruin Manchester United manager Moyes' first game against his former club.
In an open match, former Everton striker Wayne Rooney's deflected effort struck the post in the first half before substitute Gerard Deulofeu shot straight at David de Gea immediately after coming on for the Toffees in the second half.
Danny Welbeck headed against the crossbar for United after Patrice Evra's header had forced a reaction save out of Tim Howard before Kevin Mirallas curled a free-kick against the post for Everton.
Roberto Martinez's side kept the ball alive and, when Romelu Lukaku's mis-hit shot dribbled across the face of goal, Oviedo was the most alert to side-foot past De Gea and inflict a first defeat in 13 games on United.
Liverpool striker Suarez scored four and set up the other in a 5-1 thrashing of Norwich at Anfield to take his record against the Canaries to 11 goals in five appearances.
Suarez set Liverpool on their way in magnificent fashion on 15 minutes when his stunning volley from 35 yards looped over John Ruddy after Steven Gerrard had headed back the Norwich keeper's clearance.
The Uruguay international struck again on 29 minutes when he crashed home a volley from six yards out after Norwich had failed to stop Philippe Coutinho's corner from the left.
A brilliant solo effort brought about Suarez's hat-trick on 35 minutes when he lifted the ball over Leroy Fer before lashing a right-footed effort past Ruddy from the edge of the area.
Suarez capped his virtuoso performance on 74 minutes with a fourth goal as he curled an excellent 30-yard free-kick over the wall and inside the post.
Norwich pulled a goal back on 83 minutes when Bradley Johnson glanced home Nathan Redmond's corner before Suarez set up Raheem Sterling to score Liverpool's fifth on 88 minutes.
Arsenal racked up a sixth consecutive home win after goals early in each half secured a 2-0 victory over Hull City at the Emirates Stadium.
Nicklas Bendtner was handed his first Premier League start of the season and he rewarded Arsene Wenger's faith within two minutes when he headed home Carl Jenkinson's cross for his third goal in four appearances against Hull.
Arsenal's second goal arrived two minutes after half-time when Nacho Monreal kept the ball in play on the left before Aaron Ramsey's brilliantly incisive pass played in Mesut Ozil to finish past Allan McGregor.
Eden Hazard scored a goal in each half, and Phil Bardsley netted at both ends, as Chelsea got the better of Sunderland 4-3 following a topsy-turvy game at the Stadium of Light.
Sunderland took the lead on 14 minutes through Jozy Altidore's first Premier League goal for the club after the USA international striker capitalised on some poor Chelsea defending.
Chelsea, though, were level within three minutes when Frank Lampard headed home from close range after Sunderland failed to deal with a corner.
The visitors went in front nine minutes before half-time when Belgian schemer Hazard cut inside and fired a shot into the corner.
But Sunderland were quick out of the blocks in the second half and equalised five minutes after the break when Chelsea were sloppy in defending a corner and captain John O'Shea lashed home.
Chelsea regained the advantage on 62 minutes when Hazard exchanged passes with Lampard before rifling home a shot past Vito Mannone.
The Blues scored a fourth goal on 84 minutes when Bardsley turned Demba Ba's cross into his own net before the full-back scored at the right end two minutes later when he turned home Sebastian Larsson's corner.
Manchester City won away from home for just the second time this season after hanging on for a 3-2 success at West Bromwich Albion.
City belied their away day troubles to take a ninth minute lead when Sergio Aguero thumped home Pablo Zabaleta's pull-back following a slick move.
Another impressive team build-up brought City's second goal on 24 minutes which was finished when Yaya Toure converted Aleksandar Kolarov's low cross from close range.
Ivory Coast midfielder Toure notched his second goal of the game on 74 minutes when he scored from the penalty spot after Claudio Yacob had fouled Kolarov.
West Brom pulled one back with five minutes to go when Zoltan Gera's shot hit Kolarov and then City keeper Costel Pantilimon before going over the line.
And Victor Anichebe then set up a tense finale by scoring his first goal for Albion in stoppage-time, but City held on for the points.
Tottenham came from behind to lift some of the pressure on Andre Villas-Boas after claiming a 2-1 win over Fulham at Craven Cottage.
Fulham grabbed the lead on 57 minutes when former Spurs forward Dimitar Berbatov played in Ashkan Dejagah to finish past Hugo Lloris.
But Spurs equalised through Vlad Chiriches' half-volley on 73 minutes and then grabbed the points with eight minutes to go when Lewis Holtby crashed home a left-footed shot from 25 yards.
Aston Villa condemned Southampton to a third straight Premier League defeat after Fabian Delph's cracker secured a 3-2 win at St Mary's.
Villa hit Southampton on the counter-attack on 15 minutes when Maya Yoshida failed to cut out Delph's pass on the halfway line and Gabriel Agbonlahor raced clear before skipping past Dejan Lovren and placing his shot past Paulo Gazzaniga for his first goal of the season.
Southampton equalised three minutes into the second half when Jay Rodriguez headed Nathaniel Clyne's cross beyond the reach of Brad Guzan.
But Villa were back in front on 64 minutes when Karim El Ahmadi fired over a cross from the right and Libor Kozak's header had too much power for Gazzaniga.
The home side hit back again five minutes later when Rickie Lambert headed the ball back across goal and first-half substitute Dani Osvaldo stooped to nod past Guzan.
Villa took the lead for a third time with 10 minutes to go when Delph collected the ball just inside the Southampton half and ran towards goal before rocketing a 25-yard left-footed shot into the top corner.
Newcastle's four-match winning run was halted in abrupt fashion following a 3-0 defeat by Swansea at the Liberty Stadium.
Swansea were in front in first-half stoppage-time when Newcastle were unable to clear a set-piece and Nathan Dyer volleyed home.
The hosts' second goal arrived in fortunate fashion on 67 minutes when Tim Krul saved from Jonjo Shelvey, but the ball rebounded over the line off Newcastle full-back Mathieu Debuchy.
Shelvey was then on the scoresheet himself with nine minutes to go as he curled a shot past Krul after being teed up by Jonathan de Guzman.
Struggling Stoke and Cardiff played out a 0-0 draw at the Britannia Stadium, with the hosts' Geoff Cameron having a first-half goal ruled out for offside.