Suarez scored a memorable treble on his last visit to Carrow in April and he took just 67 seconds to open his account in Norfolk this season, scoring from the edge of the box to make it 1-0.
He then capitalised on a terrible mistake by Michael Turner to make it 2-0 before laying on Nuri Sahin, who made it 3-0 just after the break.
The former Ajax forward then took his tally for the season to six when he curled home from the edge of the box and Steven Gerrard scored Liverpool's fifth after Steve Morison had pulled one back for Norwich.
Substitute Grant Holt overpowered his marker to score another consolation for Norwich with three minutes to go, but there was no doubt who the bossed the game at the final whistle, as the home fans booed their team off following a poor display, while Liverpool enjoyed their first league win of the season.
A lot had been made of Liverpool's poor start to the season under Rodgers before today's game, but the Merseysiders were a class apart at Carrow Road.
Teenage trio Raheem Sterling, Suso and Andre Wisdom all performed brilliantly and Sahin proved the perfect foil to Suarez's penetrating runs through the Norwich defence.
They played with fluency, pace and brutal finishing that will give Rodgers hope that he can deliver success at Anfield during his first term in charge.
A Liverpool rout seemed on the cards from the start. With just over a minute on the clock, Glen Johnson's through-ball was cleared only as far as Suarez and he drove home from 18 yards.
Daniel Agger headed over a free-kick as Liverpool searched for a quick second, but Norwich then stormed forward through Simeon Jackson, who drew a top-drawer reaction save from Pepe Reina at his near post.
Sterling then knocked Suarez's pin-point ball across the box but Norwich cleared just before the unmarked Sahin could pull the trigger.
Suarez was at the centre of controversy in the 23rd minute when the Uruguayan raced through after a long ball over the top and broke in to the box.
Just as the forward was about to shoot Leon Barnett barged in to back of Uruguayan and brought an elbow down on his shoulder, but referee Mike Jones waved play on despite the please of the Reds players, who were infuriated at the decision.
Suarez was involved in the action again minutes later when he played Gerrard through with a wonderful 40-yard pass but John Ruddy palmed the skipper's header away from close range.
Jackson blazed over a good chance to put Norwich back in the game and Suarez made the hosts pay.
Barely 30 seconds after shooting wide while one on one with Ruddy, the Uruguayan pinched the ball off a lazy Turner, nutmegged the defender and slotted finished from 12 yards before goading the home supporters in his celebration.
After seeing his team booed off at half-time, Chris Hughton brought on Holt and the Canaries started playing much better.
Hughton's team should have pulled one back in the 46th minute when Jackson fired a cross the box with a volley but the ball cannoned off Robert Snodgrass' knee and flew over the bar from barely two yards.
Once again Norwich were made to pay for their miss as Suarez came in off the flank and unselfishly squared to Sahin, who made it 3-0. Some Norwich fans had seen enough and left even though the game was only 56 minutes old.
Andrew Surman blazed a terrible shot wide and almost predictably Liverpool went down the other end and scored.
Norwich foolishly stood off Suarez and he curled a brilliant finished wide of the outstretched Ruddy to complete his hat-trick.
The Canaries restored some pride soon after when Reina spilled Russell Martin's shot in to the path of Morison and he calmly swept home.
Liverpool underlined their dominance with little over 20 minutes left when Sterling took advantage of a slip by Javier Garrido to square for Gerrard, who scored thanks to a big deflection off Barnett that wrong-footed Ruddy.
Suarez almost bagged his fourth four minutes later when he latched on to Gerrard's pass down the left wing, but the forward could only find the side-netting.
The away fans sung the names of Suarez and Rodgers in the dying minutes, clearly delighted at their team's first win of the season.
The home fans, meanwhile, vented their frustration by booing their team off as their winless run under Hughton continued.