The notorious Uruguayan then sealed the points late on, notching his side's first second-half goal of the season after Emanuele Giaccherini's tap-in had given the hosts some hope.
Liverpool had the ball in the net moments later as Martin Skrtel powered a rebound beyond Kieren Westwood, who had made a smart save to keep out Steven Gerrard's low free kick. However, the Slovak's effort was correctly ruled out for offside.
Sunderland looked threatening at the other end and Liverpool's crossbar was rattled midway through the first half by a curling free kick from Larsson.
The home fans, who appeared to be rejuvenated following Kevin Ball's appointment as caretaker boss, then appealed for a penalty as Giaccherini went down under Kolo Toure's strong challenge. Referee Howard Webb was correct to wave away the appeals.
Webb then made another big decision by allowing Sturridge's opening goal to stand on 28 minutes, despite the Liverpool forward knocking the ball over the line with his right arm following a set piece.
Sturridge's opener ended Liverpool's four-hour run without scoring a goal, but Reds fans had to wait just another eight minutes to see their side find the net again.
Suarez tapped home from close range following a brilliant run and cross by Sturridge.
Suarez flashed a speculative volley just wide from a narrow angle soon after the restart, but, roared on by their biggest crowd of the season so far (41,415), Sunderland then got on the front foot.
Adam Johnson went close with a low effort from 25 yards, before another long-range effort yielded a goal for the Black Cats seconds later.
Ki's stinging drive was only parried by Mignolet, allowing Giaccherini to follow in and net his second goal in as many games.
Liverpool were rattled and, full of newfound belief, Sunderland piled forward, forcing Mignolet into action on numerous occasions.
However, Sunderland's frequent voyages forward created many counter-attacking opportunities for the Reds.
Jose Enrique crossed for Suarez, who could only bundle the ball into the arms of Westwood from the centre of the six-yard box.
Suarez did get the better of Westwood again with three minutes remaining. Sturridge pulled the ball back from the left byline and the Uruguayan swept into the net with his left foot.
Liverpool's second away victory of the season sees them climb to second in the Premier League table.
Meanwhile, Sunderland, who are the only top-flight side yet to register a win this season, stay bottom of the pile.