Suarez added to Daniel Sturridge's opening goal to put Brendan Rodgers's side 2-0 ahead at the break against the winless and managerless hosts, who remain bottom of the table with just a single point from six games.
Sunderland pulled one back early in the second half through Emanuele Giaccherini, but Suarez celebrated the end of his 10-game ban with a second as stoppage time approached.
It had taken him only 36 minutes to score his first goal, having made his return from his suspension for biting Chelsea's Branislav Ivanovic last season in the mid-week League Cup loss at Manchester United.
The 26-year-old raised his top as he was mobbed by his team-mates to reveal a T-shirt with a message welcoming his new-born son, Benjamin, to the world following his birth on Thursday.
More From This Section
Suarez's close-range strike doubled a lead procured by Sturridge after 28 minutes; the England forward's 11th goal in as many Premier League games.
There was a touch of controversy as the ball appeared to find the net via the Liverpool forward's elbow, as he rose unmarked to meet Gerrard's corner and record his seventh goal of the campaign.