This was the 144th meeting between the Lancashire sides, and two of the country's pioneering clubs who faced each other on the first day of the Football League in 1888.
But they have not met before in the FA Cup and it was Preston who emerged comfortably victorious to reach the quarter-finals for the first time since 1966, with fans chanting "We're going to Wembley".
Both Championship games had finished goalless this season, including the fixture two weeks ago which was paused for several minutes in the second half when Hannibal reported an accusation of racism against Osmajic to the referee.
Hannibal was left out of this game and 25-year-old Osmajic was the central figure - there seemed an air of inevitability that he would score when he screwed wide early on.
Considering the circumstances and the chants aimed by Burnley supporters in his direction, the goal Osmajic converted was extremely cool, setting himself inside the box before striking low into the corner and giving Hladky no chance to save it.
Brady's sweet opener was just the second goal Burnley had conceded since 21 December, while Keane's effort on 73 minutes meant the Clarets conceded three in a game for the first time this season.
Having made nine changes for the tie, manager Scott Parker had an eye on the Championship as they aim to make an immediate return to the Premier League.
They were often disjointed and appeared off the pace at Deepdale, rarely looking like they could trouble Woodman, with Manuel Benson blazing high and wide when attempting to create some magic.
Luca Koleosho flashed a low strike wide of the far post and had an effort cleared off the line in between the first two Preston goals, but Burnley can now concentrate on the league with a trip to Cardiff on Tuesday evening.