Chris Herd's curling shot put the visitors ahead as early as the 11th minute, as The Reds looked to be stalling at home once again.
But Suarez was the first to react after Daniel Agger’s header clattered the bar to set up a grand stand finish as Shay Given saved from both Andy Carroll and Steven Gerrard in the dying minutes.
Having scored just 19 goals all season at home, the Anfield faithful feared the worst when Herd swept home expertly to hand Villa the lead.
Liverpool rallied and Dirk Kuyt should have equalised before half time, but inexplicably blazed over from a yard out after Gerrard’s cross, with the goal gaping.
The second half saw similarly presentable chances squandered for the home side, as Villa looked to cling on to their early goal, offering little in attack.
And their negativity eventually proved to be their undoing as Gerrard’s whipped cross found the head of substitute Agger, and while the Dane could only hit the bar, Suarez swooped to knock beyond Given from a yard.
The effort was the catalyst for a late surge from the home team, which almost proved fruitful.
Carroll’s downward header was comfortably saved by Given, Gerrard’s long-range effort was beaten away and Kuyt couldn’t react quick enough when the ball flew his way in the fifth minute of injury time.
The draw eases Villa’s relegation fears, while Liverpool’s home form continues to undermine their European hopes.