Gareth Southgate's men were heading out as they trailed 1-0 in Gelsenkirchen on Sunday before Bellingham produced a stunning overhead kick to equalise in the 95th minute.
Less than a minute into extra-time, Kane then headed in his 65th international goal to take England into a quarter-final meeting with Switzerland on Saturday.
"Two superstars, Jude and Harry come up with the goods," Pickford told reporters at England's training camp in Germany on Monday.
"Jude is the golden boy again with the overhead kick to keep us in the game, keep us in the tournament. We never say die and that's what it was last night."
Despite reaching the last eight, England's performances have been heavily criticised.
Hyped as pre-tournament favourites, they have looked fatigued and short on creativity in all four games of the tournament so far.
"(A) rollercoaster but we got over the line," added Pickford on his emotions during the Slovakia match. "It doesn't matter how you play, you've got to show resilience and win the match and that's what we did."
A much-improved display will be needed against the Swiss, who are unbeaten at the tournament so far and easily eliminated holders Italy 2-0 in their last 16 clash on Saturday.
Pickford, though, cited Portugal's unlikely triumph at Euro 2016 as evidence that England do not need to be perfect to end a 58-year wait to win a major tournament.
"I think if you look at past tournaments, in 2016 Portugal drew all three group games and went onto win the tournament," said the Everton goalkeeper.
"We've just got to keep fighting, keep believing and train hard going into a tough game on Saturday."
Luke Shaw took a full part in training on Monday to offer Southgate hope he could be fit for the quarter-finals.
The Manchester United left-back has not played a minute for club or country since February due to injury.
Southgate's decision to bring him as his only natural left-back in a 26-man squad has been heavily criticised and Shaw may be rushed into action as Kieran Trippier was forced off with an injury in the second half against Slovakia.
