FA Cup: Rochdale 1 Stoke 4
The attacker broke the deadlock in the fourth minute at Spotland with a sensational volley, but went down under little pressure shortly after the half-hour mark.
Rochdale grew in confidence after Bojan had left the field, but fell two behind when the former Barcelona man's replacement, Stephen Ireland, applied the finish to a squared Victor Moses pass in the 52nd minute.
Moses added a third for Stoke nine minutes later with a fine solo effort and, although Rhys Bennett grabbed a late consolation, Rochdale conceded again in stoppage time as Jonathan Walters converted Marko Arnautovic's right-wing cross.
The fixture rounded off a remarkable cup weekend that saw both Chelsea and Manchester City eliminated by lower-league opposition, but progression rarely looked in doubt for Mark Hughes' men, who will now face the Welshman's former club Blackburn Rovers in round five.
Captain Ryan Shawcross was included in the Stoke starting XI having sufficiently recovered from a groin injury, while Walters shook off a back problem to be involved, and the Premier League outfit soon asserted their superiority.
Jack Butland's long ball forward was only headed clear as far as Bojan by Olly Lancashire, and the Spaniard found the net with a perfectly judged, dipping volley from 20 yards.
A move that started with the kind of punt so often associated with Stoke sides of the past was expertly finished by the man on whom so many of the club's fans pin their hopes.
The lively Stephen Dawson saw claims for a penalty turned down when he went down under the challenge of Stoke new boy Philipp Wollscheid, before tempers flared in the 27th minute following a full-blooded Peter Vincenti challenge on Moses.
Bojan was first to give the Channel Islander a piece of his mind, before players from both teams got involved.
That incident did not make Bojan popular with the home support, and his match came to a premature end three minutes later when he was forced off with a knee injury after going down unchallenged as he charged into the penalty area.
Bennett diverted wide from a corner as Rochdale pushed for a leveller, before Matt Done poked wide under pressure from the onrushing Butland after being sent clean through on goal by a defence-splitting pass.
Steven N'Zonzi was replaced by Steve Sidwell at half-time, although it was not immediately clear whether the midfielder had picked up an injury.
Marc Wilson's venomous long-range strike at the beginning of the second half caught an unsuspecting steward on the head, forcing the stricken stadium employee to vacate his post, but it was the Rochdale defence who were taken by surprise when substitute Ireland ghosted into the box to net Stoke's second.
Moses picked up the ball on the left and squared for Ireland – scorer of two goals from the bench at home to Wrexham in the last round - to tap home.
Nigeria international Moses then got in on the act himself, beating Joe Rafferty twice before curling in from a tight angle to put the match beyond Rochdale.
Bennett side-footed home in the 78th minute as Stoke failed to clear their lines from a free-kick, but it was scant consolation for Rochdale, who failed to add their name to the honours list on a classic FA Cup weekend.
Walters rubbed salt in the wound for the hosts with a simple finish from an Arnautovic cross at the death.