The Uruguayan, who had earlier been booked for diving, struck 19 minutes from time to cancel out Steven Fletcher's first-half opener and give manager Brendan Rodgers a little breathing space after the club's worst start to a season in half a century.
On a day when the victims of the Hillsborough disaster were at the forefront of minds - Liverpool's players walked out wearing tracksuit tops with the number "96" on the backs and flags at the Stadium of Light flew at half-mast - both Glen Johnson and Steven Gerrard had earlier hit the woodwork.
Fletcher's third goal in two league starts looked like it was going to be enough to deepen the Reds' woes to the delight of the bulk of the 41,997 crowd, but Suarez ensured Liverpool did not return home empty-handed.
At the end of an emotional week off the pitch, Liverpool arrived on Wearside hoping for much-needed points on it, but as so often during the opening weeks of the season, they headed for the dressing room at half-time frustrated.
For the best part of half an hour, they dominated possession, but were unable to make the pressure tell.
Raheem Sterling threatened repeatedly down the right, but with Sunderland debutant Danny Rose giving as good as he got, the winger's efforts went largely unrewarded.
It started in promising fashion for the Merseysiders as Jonjo Shelvey fired just wide with four minutes gone after combining well with Suarez, who then also missed the target seven minutes later.
Shelvey played striker Fabio Borini into space on the right with a well-weighted 13th-minute pass, and his cross was deflected goal-wards, but straight to keeper Simon Mignolet.
The Belgian came to the Black Cats' rescue five minutes later when Borini ran on to Carlos Cuellar's woefully weak header and fired in a shot from a tight angle which the keeper managed to block.
Sunderland almost took the lead when they were awarded a 20th-minute free-kick which Sebastian Larsson curled into the box and just wide of the far post with Pepe Reina flat-footed and no one able to get a touch.
The game changed in an instant with 29 minutes gone when Sunderland full-back Craig Gardner slipped all too easily past Johnson and crossed for Fletcher to get in front of his man and turn the ball home from close range.
Liverpool's confidence visibly drained away as the home side started to take control, and the response from Rodgers' men was tepid.
Their afternoon took a turn for the worse when Suarez surged into the box eight minutes before the break and collapsed under John O'Shea's challenge, earning a yellow card rather than the penalty for which he was hoping.
Cuellar failed to re-appear after the break and was replaced by Titus Bramble, and the newcomer found his defence stretched within two minutes when Gerrard and Suarez combined to set up Shelvey, whose skidding drive was saved.
The visitors had once again started confidently and Johnson deserved better when he sent an inviting ball across the face of goal, but found no-one there to meet it.
Johnson opted to go it alone with 51 minutes gone when he cut inside and went right-footed for the top corner, only to see his effort come back off the crossbar with Mignolet beaten.
The Reds were desperately unfortunate not to be back on level terms on the hour when, after Mignolet had beaten away Sterling's cross, Shelvey squared for Gerrard to clip the outside of the post with a guided attempt.
Rodgers replaced Borini with winger Stewart Downing four minutes later, but it was defender Martin Skrtel who went close after Gerrard's deflected shot sat up nicely for him, but his header was well-saved by Mignolet.
Liverpool eventually made the breakthrough with 71 minutes gone when Sterling rounded Rose to cross from the right and Bramble could only block the ball to Suarez, who gleefully smashed it past the keeper from close range.
Shelvey might have won it 10 minutes from time when he cut inside on to his left foot and went for goal, but Mignolet got down well to save and Suarez could not climb high enough to head the rebound on goal.