The Uruguay international won a contentious second-half penalty which Steven Gerrard converted to make it 2-2, despite goalkeeper Brad Guzan and his Aston Villa team-mates protesting he had barely touched the striker.
After nine wins and only one defeat at home, Liverpool had found themselves trailing to goals from Andreas Weimann and Christian Benteke inside 36 minutes.
Liverpool's principal owner John W Henry, watching his first home game since appointing Brendan Rodgers in the summer of 2012, must have wondered quite what was going on. However, Daniel Sturridge pulled a goal back on the stroke of half-time before Gerrard equalised soon after the break.
For the first time in eight home league matches, 22-goal striker Suarez failed to score - missing the chance to equal Roger Hunt and Fernando Torres' club record - but that was not Liverpool's problem.
Rodgers' selection, dropping defensive midfielder Lucas Leiva to accommodate Sturridge's first start since November 9, looked a bold one.
With Gerrard and Jordan Henderson in midfield, Sturridge, Suarez, Raheem Sterling and Philippe Coutinho were given licence to attack in an ambitious 4-2-4 formation which left Liverpool looking vulnerable. The portents were not good from the moment Gabriel Agbonlahor stabbed wide when through after just 48 seconds, with Ashley Westwood shooting straight at Simon Mignolet soon afterwards.
Villa almost went ahead when Clark lost Kolo Toure at a corner but headed against the post. That warning was not heeded, though, as Villa's opener highlighted all that was wrong with Liverpool's set-up. Agbonlahor raced down the left and Weimann cantered forward the best part of 60 yards virtually unnoticed to get on the end of the cross at the far post.
After switching to a back three, the hosts enjoyed a brief moment of respite as Sterling fired an angled shot at Guzan, but more holes were picked in Liverpool's defence for Villa's second. Agbonlahor exploited space on the right and when Mignolet and Glen Johnson both got in the way of each other trying to reach his cross, Benteke headed into an empty net.
The half was well into added time when the home side carved out their only genuine chance and it resulted in a very timely goal. Suarez picked up the ball on the edge of the area and threaded a pass through to Henderson, whose back-heel into the path of Sturridge ended with a classy chipped finish.
Change was inevitable for the second half, as Lucas replaced his compatriot Coutinho. The freedom now afforded to Gerrard was evident and his raking pass perfectly picked out Suarez, who skipped into the area and was brought down by Guzan.
Contact looked minimal, especially considering some brutal treatment dished out to the Uruguayan in the first half, but referee Jon Moss was in no doubt and Gerrard expertly converted from the spot.
With their tails up, Henderson drilled in a low shot which Guzan did well to parry low to his right while Suarez was inches wide with a 35-yard free-kick, but Liverpool eventually ran out of ideas against doughty opponents.