The cup upset - the biggest so far of the latest round - began after Jordan Willis broke the deadlock 24 minutes in.
Charlie Adam netted the equaliser from the penalty spot - his first goal since 2015 - but Jack Grimmer's strike, which beat Stoke goalkeeper Jack Butland to the bottom corner, ensured Coventry's progression.
"I'm proud. Regardless of the current troubles at Stoke, and the pressure on them, we equipped ourselves really well," Coventry boss Robins told reporters. "We went toe to toe with them. We could have been 3-0 up in the first half.
"They put pressure on us in the second half were able to bring on a wealth of experience from their bench. But then we got the second on the counter. The boys threw themselves at everything and we're in the hat for the next round.
"They were desperate for the win, as we were. There were times where you have to forgive the lads who are 18 and 19. There were so many things that were pleasing out there today."
Elsewhere in the FA Cup, Manchester City eased past Burnley while Bournemouth salvaged a draw against League One side Wigan Athletic.