Roberto Martinez hailed Phil Jagielka's equaliser in the Merseyside derby as the best goal he has ever seen live.
Everton looked set to slump to another defeat at Anfield in a match edged by Liverpool, but captain Jagielka stepped forward in injury time to drill a 25-yard shot into the top corner and earn a 1-1 draw.
It was the centre-back's first goal for the Toffees since April 2013 and an uplifting way to finish a testing afternoon for the visitors.
Everton's last victory at Anfield came exactly 15 years ago, while their wait for a derby win will now extend into a fifth year.
Martinez said: "I don't think we played the way we can on the ball. It was a real test of character. First of all you're coming to Anfield and we all know that for 15 years we've never won the game.
"In the first half we looked very much in control at times. We didn't make the right decisions in our counter-attacking situations or in the final third. I think Kevin Mirallas getting injured affected that.
"When the goal from the free-kick goes in, which I thought was a bit unlucky because it should have been a free-kick the other way, we had two choices - just accepting defeat and accepting it wasn't our day or keep fighting.
"And, with a lot of tired legs, we had an incredible belief, and I think we got the result out of that. It's the most pleasing result that we've had this season because in many ways that's going to help us towards the future.
"Last season we came here, we played the football. Technically it was the best performance that we put in away from home and we lost 4-0.
"Today we were back to our normal selves off the ball with a real intensity. And then you see someone like Phil Jagielka taking responsibility, someone that has got a lot of unfair criticism from the outside.
"That's a real example of how to face adversity and nobody deserves more than him the feeling of scoring such a special goal in front of the Kop.
"I don't think I've seen a better strike in live football. It's a very special moment in our season that could change the dynamics completely."
The main negative for Everton was a first-half injury to Mirallas, who pulled up clutching his hamstring and had to be carried off.
Both teams have had difficult starts to the season, but Liverpool looked to seize the moment as soon as the whistle went and put Everton under huge pressure.
They finally made the breakthrough when Steven Gerrard scored a free-kick in the 65th minute, and two minutes later Mario Balotelli should have doubled the advantage but hit the bar from close range.
Reds boss Brendan Rodgers, who was unhappy Gareth Barry was not penalised for an early handball in the area, said: "We didn't get what we deserved. "I think it was very clear we were much the better side. When you concede a goal like that so late on, it probably epitomised a wee bit of the luck that went against us today.
"But I'm certainly very proud of the players, I thought the intensity was getting back up to near where we've been for 18 months, tactically I thought we were very good. Any moments to defend, we defended very well.
"We took the lead, should have had arguably at least two more and should have very definitely had a penalty. For us it was disappointing, but the performance was at a much better level."
Gerrard has now scored a record nine Premier League goals in Merseyside derbies and the strike was a timely response to critics who have claimed his influence is waning.
"He's such a selfless guy Stevie and he was letting Mario take a few (free-kicks) in the first half," said Rodgers. "I was screaming for him to take it because he's got so much quality."
By Eleanor Crooks