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, the Reds 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 tasked with holding the midfield, it appeared Sturridge, Suarez, Raheem Sterling and Philippe Coutinho were given licence to attack.
It looked ambitious as what was effectively a 4-2-4 formation left Liverpool vulnerable to being dominated centrally.
The portents were not good from the moment Gabriel Agbonlahor stabbed wide after just 48 seconds, with Ashley Westwood shooting straight at Simon Mignolet soon after.
Villa seized what appeared an open invitation to dominate midfield and swarmed forward, giving their opponents no time on the ball.
Suarez, with 45 goals in his previous 49 league games, had probably never endured a half so bereft of personal chances at Anfield.
He was also treated with contempt by Villa's backline and was twice left in a heap after Ciaran Clark's knee caught him in the back of the thigh and Leandro Bacuna's elbow connected with his head.
Villa almost went ahead when Clark lost Kolo Toure at a corner but headed against the post.
That warning was not heeded as Villa's opener highlighted all that was wrong with Liverpool's set-up.
As Agbonlahor raced down the left, Weimann cantered forward the best part of 60 yards virtually unnoticed to get on the end of the cross at the far post.
A tactical change saw an immediate switch to a back-three, with Glen Johnson employed as a makeshift centre-half with Sterling and Aly Cissokho wing-backs.
It produced a brief moment of positivity as Sterling fired an angled shot at Guzan, but more holes were picked in Liverpool's defence for Villa's second.
Agbonlahor exploited the space behind the newly-advanced Cissokho, and when Mignolet and Johnson both got in the way of each other trying to reach the 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 produced only one outcome.
Change was inevitable for the second half, and while there would have been no reason not to replace the under-performing Gerrard with Lucas, it would have been a bold one so Coutinho was sacrificed.
With the back four reinstated and a defensive screen placed in front, the impact was immediate as Gerrard and Henderson both moved closer to Liverpool's main weapons Suarez and Sturridge.
The freedom now afforded to the Reds and England captain allowed him more space to operate in and, having dropped deep, his raking pass perfectly picked out Suarez, who skipped into the area and was brought down by Guzan.
Contact looked minimal, especially considering what the Uruguay international had sustained in the first half, but referee Jon Moss was in no doubt and Gerrard expertly converted from the spot.
It was his fifth goal of the season and his 12th league goal against Villa - he has scored more against them than any other top-flight side - and drew him level with Robbie Fowler on that tally against the midlanders.
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.
Frustrations began to grow and Sturridge was lucky not to be punished for lashing out at Bacuna as Liverpool eventually ran out of ideas against doughty opponents.