After losing their first three Premier League matches without scoring, the Toffees were staring down the barrel of a humiliating defeat in South Yorkshire.
Doncaster dominated the first half and took a deserved lead through a flicked header by Joe Ironside, who avoided an offside call and sparked wild scenes at the Eco-Power Stadium.
Everton held on and belatedly showed their quality, with half-time substitute Beto levelling intelligently a day after signing, before Arnaut Danjuma wrapped up a 2-1 second-round win.
“Fair play to Doncaster, they took it on in the right way,” boss Dyche said after the Toffees earned a third-round trip to Aston Villa.
“They know that there’s no pressure on them – a free hit at us and all the noise and all the rest of it and they used it wisely.
“We weren’t at the races first half but I must say the goal is three yards offside. I would expect that to be given (offside) but it didn’t.
“It’s a big learning curve for some of the younger players because that’s what it’s like to play for Everton Football Club.
“A lot of expectation regardless, no-one cares about injuries and stretched squads. I say it because I mean it and it’s true, but no one really cares, so that’s a big part of their development.
“It can’t always be rosy, it’s tough and they’re young and they’re learning.
“Second half we put more experienced players on who played very well, I thought, and made a big difference to the performance.
“I don’t think it was as good a performance as the weekend (in the loss to Wolves) but you win a game and that was important.”
Everton return to South Yorkshire on Saturday lunchtime to face Sheffield United, where towering striker Beto will surely lead the line after his man-of-the-match display.
“He’s only got here yesterday, so it’s a lot to ask, really,” Dyche said of the big-money signing from Udinese.
“We only got his clearance this morning, so we’d already set the team up. I had it in my mind to put him on at half-time regardless.
“He’s adapted very quickly, done very well tonight. With all due respect, the Premier League is different, but he’s shown the rawness, the pace and the effect that he can have on a team.
“It gives us something different, which is what we brought him here for.”
Doncaster boss Grant McCann was aggrieved his side did not get a penalty for a Vitaliy Mykolenko handball but his overriding emotion was pride after Rovers pushed Everton close.
“I’m pleased, proud of the boys’ performance,” the League Two strugglers’ manager said. “The levels were good against a top-class team.
“I thought particularly first half we were excellent in terms of what we did.
“The second half was difficult with the changes they made, they seemed to get a lot, lot stronger.
“I think we can see over the last couple of games that we’re definitely improving and we’re only going to get stronger.”