Steven Fletcher gave the home side the lead in the 29th minute against the run of play, tapping in from close range after a run and right-wing cross from Craig Gardner.
Glen Johnson smacked the crossbar in the early stages of the second half as the Reds sought a deserved equalizer.
And the leveller did finally arrive in the 71st minute, with Suarez firing beyond Simon Mignolet after Raheem Sterling's cross had caused panic in the Sunderland box.
The Reds dominated the final stages as they went in search of a winner, but the sustained spell of pressure couldn't yield another goal and they were forced to settle for a point.
The release of the Hillsborough Independent Panel's report on Wednesday made for a momentous week on Merseyside, as the 23-year quest for justice took a huge step forward.
Prior to kick-off, Liverpool's players took to the field with the number 96 printed on the back of their anthem jackets in tribute to those who lost their lives in the disaster, as well as the families, survivors and fans who have campaigned for so long.
Sunderland too showed its support by flying the flags at the Stadium of Light at half-mast in light of events this week, while also paying respects with a message displayed on the two big screens inside the ground as the game got underway.
On the field, it was Liverpool's first game in a fortnight following the international break, and Brendan Rodgers made two changes to the side that started against Arsenal last time out with Martin Kelly and Jonjo Shelvey coming in for Jose Enrique and Nuri Sahin.
Meanwhile, Oussama Assaidi was named on the substitutes' bench for the first time since he joined the club last month after recovering from a knock.
The Reds went into the game seeking their first Barclays Premier League win of the campaign at their fourth attempt — and Shelvey went close inside four minutes, dragging wide from the edge of the box after a well-worked move.
Next, Suarez spun into space 25 yards from goal before seeing a similar effort evade the far post.
In the 18th minute, Fabio Borini pounced on a weak back-header from Carlos Cuellar but could only prod into the midriff of the onrushing Mignolet on the stretch.
Liverpool were by no means in complete control, but it was to threaten again soon after. Another finely executed passage of play saw Shelvey slip a neat reverse ball into the path of Suarez's run into the box. The Uruguayan produced a first-time lay off to set up Borini on the edge of the box, and his snap-shot was beaten away by Mignolet.
And the Reds were left to rue their early missed opportunities on 29 minutes when Sunderland took the lead with its first effort on target of the afternoon.
Gardner skipped beyond two challenges on Liverpool's left before delivering a teasing centre that evaded the hand of Pepe Reina and allowed Fletcher to tap in from close range.
Liverpool continued to probe, though it was unable to find the breakthrough that would ensure the teams went into the break on level terms.
A positive start to the second half was the order of the day for the visitors, and after Shelvey had tested Mignolet from distance, Johnson cannoned a curler against the crossbar after collecting possession in space down the left.
A spell of Sunderland pressure followed, but Liverpool players were left holding their heads in their hands on the hour when Steven Gerrard shaved the base of the post with a side-footer from Shelvey's tee-up after Sterling's deep cross had initially caused panic in the home side's defence.
The chances were now starting to come more frequently and Skrtel's glance goalwards from a deflected Gerrard strike almost caught Mignolet off guard, but a lack of power meant the 'keeper was able to collect.
Then, in the 71st minute, Sterling's ball to Suarez created a shooting opportunity for the No. 7, however his attempt was deflected away into the path of Johnson, who drilled over the bar.
Sterling's influence on the contest was growing — and it was, therefore, no surprise he played a key role in Liverpool's equaliser a minute later.
The 17-year-old danced into space beyond Danny Rose before feeding the ball into the danger zone. Titus Bramble was unable to clear, allowing Suarez to slam home with aplomb.
It was no more than Liverpool deserved and with time left for a winner, Rodgers' side went in search of all three points, with Shelvey's jinking run and strike thwarted by a fine stop from Mignolet.
The Reds continued to press until the final whistle, but were unable to muster the goal their display merited.
Liverpoolfc.com journalists' man of the match: Raheem Sterling.
Photo via Liverpoolfc.com