Having completed a 10-game ban for biting Chelsea’s Branislav Ivanovic last season, Suarez scored in each half at the Stadium of Light as Liverpool ended a run of three games without victory.
Daniel Sturridge had given Liverpool the lead with his fifth goal of the season, with Emanuele Giaccherini replying for Sunderland early in the second half after Suarez had made it 2-0.
“He’s getting the reward for his hard work over the period that he was off,” said Liverpool manager Brendan Rodgers of Suarez, who made his comeback in the mid-week League Cup loss to Manchester United.
“I thought he was excellent the other night at Old Trafford, but to come in and get two goals…
“Him and Daniel up front, they’re a real, real handful. We’ve changed our system a wee bit to suit the players that we have and it works reasonably well for us, the 3-4-1-2.
“Those two at the tip of the structure are exceptional. They combine very well.”
It was Sunderland’s first league game since the dismissal of manager Paolo Di Canio, but although they are now four points from safety, interim coach Kevin Ball took heart from their second-half display.
“We are disappointed with the result, but in the second half it was a fantastic reaction to going 2-0 down,” he said.
“I didn’t think 3-1 was a fair reflection of the game. If you look at today as a whole, I think we put in a good performance.”
Both sides threatened from early set-pieces, Keiren Westwood denying Liverpool captain Steven Gerrard and Sebastian Larsson rattling the visitors’ bar, before the away side went ahead in contentious circumstances in the 28th minute.
Gerrard’s left-wing corner found Sturridge unmarked at the back post and although he shaped to head the ball, it actually struck his elbow before bouncing over the line.
Sunderland’s defenders did not appeal, however, and eight minutes later Sturridge’s teasing low cross from the right allowed Suarez to slide home his first goal since completing his suspension.
The Uruguayan celebrated by lifting his jersey to reveal a T-shirt bearing a picture of his family with his new-born son Benjamin, who was born on Thursday.
The hosts were quick to react though, and after former Sunderland goalkeeper Simon Mignolet had saved with his feet from Craig Gardner and Adam Johnson had shot narrowly over, they drew level in the 52nd minute.
Mignolet could only parry a low drive by Ki Sung-yueng and Giaccherini swept the loose ball home with his left foot to reduce the arrears.
There were chances for both teams, Mignolet denying Gardner again, while Westwood saved from Victor Moses and Kolo Toure, before Suarez tucked in his second goal from another Sturridge cross in the 89th minute.
Earlier, Norwich City won 1-0 against Stoke City at the Britannia Stadium to register their second victory of the league season and only their second away win of 2013.
Stoke manager Mark Hughes has been praised for introducing a more possession-oriented approach since succeeding Tony Pulis in the close season, but his side produced a strangely subdued display.
Jonathan Howson claimed the game’s only goal in the 34th minute with a low drive from 25 yards that dipped right in front of Stoke goalkeeper Asmir Begovic.
The two teams now have identical records after seven games of the season and sit three points above the relegation places.
“This is a tough place to come. We had really good spells where we dominated possession,” said Norwich manager Chris Hughton.
“It’s about picking up points and hopefully this will give us some confidence.”
Hughes was aggrieved that Stoke were denied a penalty after Leroy Fer appeared to hold back Kenwyne Jones early in the second half.
“If the referee sees it, he will agree there is an infringement,” said the Welshman.
“Kenwyne Jones was trying to get on the end of the ball; the letter of the law says it should be a penalty.”
Arsenal moved two points clear at the top of the table following a 2-1 win at Swansea City on Saturday, while Manchester City lost 3-2 at Aston Villa and Manchester United were stunned 2-1 by West Bromwich Albion at Old Trafford.