The visitors twice hit the woodwork at the Stadium of Light but, despite dominating large periods of the game, Brendan Rodgers' side remains without a Premier League win.
The Reds remained frustrated as they continue to look for their first win of the Premier League season with four matches played.
The Black Cats will also be unhappy that they were unable to notch their first three points of the campaign in the Stadium of Light clash.
Jonjo Shelvey had the first shot of the match, snatching a 25-yard strike wide. Luis Suarez fired an identical shot wide of goalkeeper Simon Mignolet’s right post a few minutes later.
Center back Carlos Cuellar caused Sunderland fans to squirm when he tried to head back to the shot-stopper but his headed pass lacked power, and Fabio Borini almost put Liverpool ahead but Mignolet saved smartly and Shelvey fired the rebound wide.
With 29 minutes on the clock Sunderland took the lead with essentially its first shot of the match. Craig Gardner did the hard graft, dribbling down the right before firing a pinpoint cross for teammate Fletcher to tap in from six yards out.
Liverpool seemed rattled by going behind against the run of play. Suarez, who was already being booed whenever in possession, added to the away side’s frustration by being deservedly booked for diving inside the penalty area.
The second half began with the visitors back in the ascendancy, Johnson rattled the crossbar after cutting inside onto his favored right-foot from 20 yards.
Raheem Stirling - still being favored on the wing ahead of Stewart Downing - sent in a dangerous ball from the right which was semi-cleared, allowing Steven Gerrard to gleefully latch onto, then less gleefully fire inches wide, skimming the post on its way off the pitch.
With 71 minutes gone Sterling’s efforts were finally rewarded. The winger sent in another tricky cross from the right which ricocheted off Titus Bramble’s, allowing Suarez to pounce and tap in to level the scoreline.
The Uruguay international had a chance to put his team ahead as the game dragged on, but he couldn't nod Shelvey’s saved shot on target, and the home fans were relieved to watch the striker’s effort sail well over.
There was not to be a late winner at the Stadium of Light, meaning both clubs remain winless in the bottom half of the Premier League table.