Rovers’ only victory of the season came in the first round when they beat League Two Walsall on penalties.
Goals from Tom Ince and in-form Sam Surridge did the damage at the Bet365 Stadium.
Manager Richie Wellens named eight changes from the side which secured a goalless draw against Portsmouth on Saturday. And they were forced into an early change when Cameron John went off injured after just 13 minutes.
The game started quietly before Surridge gave the visitors a warning sign when he fired wide. Shot-stopper Louis Jones produced a good save to keep out Surridge’s header.
Jones did well to keep out Morgan Fox’s long-range effort, but Ince pounced to tap home the opener in the 39th minute.
Rovers’ first real chance came when substitute Charlie Seaman was denied by Adam Davies.
The visitors were broken down again just after the break as Surridge got his deserved goal when he converted Alfie Doughty’s cross in the 49th minute.
Wellens’s side pushed to get themselves back into the game after going two behind.
Seaman squandered another chance when he blasted over the bar before Aidan Barlow also failed to hit the target.
Down the other end Jones did well to keep out Stoke substitute Jacob Brown.
Stoke: Davies, Chester, Batth, Fox, Doughty, Porter (Vrancic, 73), Sawyers, Thompson, Ince, Norton (Brown, 66), Surridge (Fletcher, 66). Unused substitutes: Bonham, Tyman, Østigard, Clucas.
Doncaster: Jones, Knoyle, John (Seaman, 13), Blythe, Horton, Greaves, Close (Rowe, 45), Smith (Harrison, 81), Barlow, Gardner, Williams. Unused substitutes: Dahlberg, Williams, Ravenhill, Galbraith.
Referee: M Edwards (Tyne & Wear).