The Uruguay international, whom Everton manager David Moyes claimed should have been sent off for a foul on Sylvain Distin, poked in Sebastian Coates' header in added time only to be denied by an incorrect linesman's flag for offside.
Suarez's shot, deflected off Leighton Baines, produced the opening goal and celebration of the season so far as he ran and dived full-length in front of Moyes in response to the Everton manager's pre-match comments about him going to ground easily.
The Liverpool striker then glanced in a header to put his side 2-0 up inside 20 minutes only for Leon Osman and Steven Naismith to respond before half-time.
In the second half Suarez was booked for standing on Distin's Achilles tendon after the ball had gone before being denied a dramatic winner.
"I thought it was great. I actually quite liked that - it is the sort of thing I would have done had I scored," said Moyes of Suarez's celebration.
"But I think he was a lucky man to stay on the pitch with his tackle.
"The one down the back of his (Distin's) leg - he's topped him, definitely."
Brendan Rodgers, in his first Merseyside derby, disagreed.
"I didn't think it was a red card," he said.
"Sometimes you get that with strikers, sometimes it is just a fraction and he is trying to time the ball as he's playing it."
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The managers, however, were in agreement in relation to Suarez's disallowed goal.
"We're so disappointed that it didn't count because for me it's an easy decision," said Rodgers, who has regularly complained about his side not getting decisions from referees.
"I'm not going to keep going on about officials because it's been happening to us all season.
"The ball's in the air for a long time, the official's just looking across the line so he's clearly onside and when Coates heads it Luis is minimum level and it's a wonderful finish.
"He should have had a hat-trick and it would have been a brilliant win for us."
Rodgers also complained about Everton's equaliser, adding: "If you watch any of the replays and David even admitted it himself in the game, it was our throw.
"They took the throw, go up and score but it has gone so I'm not going to get too heated over it."
Moyes admitted the linesman made the wrong call for Suarez's late strike, although he argued Liverpool should not have been awarded the free-kick which led to it.
"It was a goal - but I didn't think it was a foul on Gerrard in the middle of the pitch," he said.
"That was the wrong decision but it was the wrong decision not to give the goal."
Ironically, considering Moyes' comments about Suarez, Everton captain Phil Neville was booked for diving in the first half.
Moyes refused to enter the debate, saying "Just don't start me", but to his credit pointed out his player's error.
"I said at half-time it was completely wrong, you don't do that and hopefully he won't do it again," said the Scot.
Neville admitted his culpability by telling Sky Sports 1: "It was a stupid thing to do.
"I don't often find myself at that end of the pitch and I actually thought Daniel Agger was coming to take me out.
"Rightly so I got my butt kicked at half-time, put myself under pressure with the yellow card and I couldn't really commit to tackles in the second half.
"I won't do it again, that's for sure."