Rochdale exited the Carabao Cup at the second-round stage after defeat at Premier League Stoke City.
The home side scored three times in the first half to all but seal their progress, with two goals from Joe Allen and one from Peter Crouch.
Dale rallied after the break but Ramadan’s goal ten minutes from time completed the scoring for The Potters.
There were four changes to the side from Saturday, as Matty Done, Keith Keane, Andy Cannon and Mark Kitching all came into the side.
Stoke’s side was packed full of Premier League and Champions League experience, and it was former Barcelona man Bojan who tested Brendan Moore first with a shot from range.
Moore stuck out a strong hand to deny Josh Tymon from the edge of the area after he had broken through the middle, but there was nothing he could do to deny the opener.
Erik Pieters cross from the right side was met by the head of Allen six yards out, whose effort deflected off Harrison McGahey into the back of the net.
Ramadan fired a first-time strike just wide after Crouch’s knock down from the edge of the area, whilst Done had Dale’s first effort on target when he headed goalwards after a clearance had looped skywards, but Lee Grant collected.
Allen turned provider for Stoke’s second just before the half hour mark, when he squared the ball from the right side of the area to Crouch, who had a simple tap-in from the edge of the six-yard box.
The Potters added a third before the break, when Pieters supplied Allen again, this time with a square ball across the ground which Allen converted for his second.
Steven Davies replaced Keane at the break, and a clearance from a Done cross by Bruno Martins-Indi hit the striker before flashing narrowly wide just moments after the restart.
Bradden Inman was replaced by Jordan Williams eight minutes into the second half with Dale’s second change, shortly before first substitute Davies forced Grant into a save low to his left with an effort from the corner of the area.
Martins-Indi created himself a good chance for Stoke’s fourth when he broke into a Dale area after a powerful surge forward, but put his effort just wide of Moore’s left post.
Charlie Adam’s superb pass forward found the run of Tymon, who saw Moore off his line and opted to chip, with his effort striking the top of the crossbar.
Pieters and Ramadan both fired over, before Bojan struck an effort just wide of the far post from a tight angle.
Cannon showed quick feet to get past a couple of Stoke defenders on the edge of the area before firing goalwards, which Grant had to collect at the second attempt, before Adam struck a sweet volley from distance which was straight down the throat of Moore.
Stoke’s fourth came ten minutes from time, when Ramadan showed great pace to get away down the right side, before he rounded Moore and slotted in from a tight angle.
Done was replaced by first-year professional Matty Gillam immediately after the goal, and he almost provided an assist for Davies, who fired just wide from the edge of the area after being played in by the youngster.
Cannon struck a half volley over, before a late effort from Ian Henderson tested Grant with the final act of the game as Dale finished strongly.