Having twice led through Philippe Coutinho and a brilliant Luis Suarez free-kick - either side of Kevin Mirallas' equaliser - Romelu Lukaku scored twice to give the Toffees a 3-2 lead with eight minutes to go.
Mignolet's Belgium team-mate appeared to have delivered a crushing blow only for Sturridge to snatch back a point in the 89th minute.
"I have to be honest once they scored at 3-2 you think you are losing the game but looking back we are always there scoring goals," he told Press Association Sport.
"We always get a chance with Daniel and Luis and we know they have goals.
"Daniel scored the equaliser which was very important for us as it means we can keep going in the way we started this season and make sure we build on that.
"If we win next week this can be a very important point for us."
Mignolet was Liverpool's best player with a string of saves in the second half as Everton pushed, initially, for their equaliser.
However, even he could not prevent the Reds' weakness at set-pieces being exploited as with the first coming from a free-kick and the third a corner.
"As a goalkeeper you don't want to be busy. You like it in training but you have to do the things you are asked and do the best you can," he added.
"It is not nice to concede from set-pieces but we also scored from set-pieces (all three goals).
"We have to make sure we do things better in the next few games.
"But Romelu is a physical presence and sometimes you have to give credit to the attacker as well."
Aside from the overall action in the game the major talking point was Kevin Mirallas' knee-high challenge on Suarez which left the Uruguay striker limping heavily for 10 minutes before half-time.
The 26-year-old was still limping when he left Goodison Park but the tackle which manager Brendan Rodgers described as potentially career-ending was not enough to prevent him completing 90 minutes and almost snatching victory when his superb effort in added time was batted away by goalkeeper Tim Howard.
Suarez, after a 7,000-mile trip from his homeland following a World Cup qualifying play-off on Wednesday, was floored for some time by Mirallas' challenge - which referee Phil Dowd decided was worthy only of a booking.
Rodgers admitted he feared the worst.
"You don't see him do that. Sometimes he rolls about but then gets up quickly but I knew it was reasonably bad when he stayed down that long," said the Reds boss.
"I don't think there are many (who could play on). His will and determination to play is unbelievable.
"He is just remarkable. He got back late Thursday night and looked surprisingly fresh in Friday's training.
"Having only one striker for six months last year gave me an idea of what his mentality is like and I know even if he is not playing well at the end of the game he can still get you something and he nearly did.
"He is a player who goes to the limit every game so I have no doubts."
Despite his disappointment at conceding three goals Rodgers was pleased with his team's battling qualities as they picked themselves up to get a draw.
"When you come off after a game like that against your rivals and play like we did in parts of the game and show the character and mentality to come from behind it tells me everything about where my group is at," he added.
"We are disappointed as well because we were 2-1 up and we should have gone 3-1 (when Joe Allen inexplicably missed the target with only the goalkeeper to beat), which would have allowed us to control the game.
"They ended up getting the equaliser and then we were not strong enough on a corner, Lukaku gets his header and it's 3-2.
"In a game we should never have been behind in against our rivals we had to show our character and that is what we did.
"A point away from home against a rival is always a very good result but obviously we will look at it and think with a bit better control we could have won."