The Premier League side’s early-season woes were intensifIed when the hosts, bottom of the EFL, took the lead just before the break courtesy of Joe Ironside.
The goal had an element of controversy with the striker looking offside from Tommy Rowe’s cross. But there was nothing dubious about the merit of Rovers’ lead, with Grant McCann producing an outstanding display, especially in the first half.
Everton made three interval changes with the arrival of £30m signing Beto, with the Portuguese striker having joined from Udinese, changing the complexion of the game in particular.
He slotted home a classy and relieving leveller on 73 minutes and pressure later resulted in a winner for Arnaut Danjuma.
Dyche spared a thought for Rovers when reflecting on the game afterwards.
He said: “Fair play to Doncaster, they took it on in the right way.
"They know there was no pressure on them and it was a free hit on us and all the noise and all the rest of it. They used it wisely, I thought.
"We weren’t at the races in the first half, but I must say, their goal was three yards offside. I’d expect that to not be given, but it didn’t (not be).
"It’s a big learning curve for some of the younger players as that’s what it’s like to play for Everton. There’s a lot of expectation.”
Offering his take, Rovers chief Grant McCann said: “I was pleased and proud of the boys. I thought the levels were good.
"We played against a top-class team and particularly in the first half, I thought we were excellent in terms of what we did.
"In the second half, with the changes they made, they seemed to get a lot, lot stronger. But right up until they scored, I thought we were really good defensively. We were strong and the duels were good and we were calm in our play.
"But the first goal was a sickener for us. From their point of view, it was an excellent goal. The second one was avoidable for us, though.
"But my overriding thoughts at the minute were that I was really pleased with the lads, they went toe to toe with a very good side.”