Liverpool broke the deadlock with a controversial goal in the first half as the ball went in off Daniel Sturridge's arm, before Suarez quickly doubled their lead.
Emanuele Giaccherini reduced Sunderland's arrears on 52 minutes but, despite a spirited performance, Kevin Ball's men could not find an equaliser and Suarez netted again in the closing stages to lift Liverpool to within two points of leaders Arsenal.
Suarez, returning to Premier League action following a 10-match ban, showed that he had not lost any sharpness with an early run and powerful shot just wide, while Martin Skrtel also had a goal disallowed inside the opening 10 minutes for offside after Keiren Westwood had brilliantly kept out a Steven Gerrard free-kick.
Sunderland played their part in a lively first 20 minutes, though, and there was a vibrant atmosphere at the ground as the home players showed an intensity which belied their standing at the foot of the table.
Sebastian Larsson struck the crossbar with a curling free-kick and Giaccherini had a penalty appeal turned down following a challenge by Kolo Toure, before Liverpool took the lead in fortuitous fashion on 28 minutes.
Sturridge leant in for a header from Gerrard's corner but ended up knocking the ball in with his arm and the incident was not spotted by referee Howard Webb or his assistant.
Sturridge turned provider eight minutes later as he collected a sublime Gerrard pass and beat his defender for pace before squaring an inviting cross across the six-yard box for Suarez to add the simplest of finishes at the far post and make it 2-0 to Liverpool.
Liverpool goalkeeper Simon Mignolet, making his first return to Wearside since a summer transfer from Sunderland, had a couple of anxious moments but did make a tremendous save just before the interval, rushing off his line to deny Craig Gardner with an outstretched right leg.
Mignolet could perhaps have done better early in the second period, however, as he could only parry Ki Sung-Yueng's low shot from 20 yards straight back out to Giaccherini and the Italian made no mistake from the rebound to pull a goal back for Sunderland.
It was almost level moments later as Mignolet reacted quickly to get down to his left and get the slightest of touches to a Gardner strike from the edge of the area.
Jozy Altidore also hit a ferocious effort from distance straight at Mignolet as Sunderland enjoyed a spell of sustained pressure, before Liverpool started to re-emerge as an attacking force and the game ebbed and flowed in an open finale.
A rasping 20-yard drive from Victor Moses was tipped over the bar by a strong one-handed stop by Westwood, while the Sunderland keeper also saved well from Toure and Suarez fired wide from the edge of the area.
The crowd spurred Sunderland on in the final 10 minutes and there were some tense times at the back for Liverpool before Suarez finally settled any nerves in the 89th minute with another finish from close range after good work by strike partner Sturridge on the break.