West Brom had taken only five points from the last possible 15 in the Championship but they had no problems in despatching managerless Coventry to proceed to the last eight for the first time since 2002.
Coventry, who sacked Iain Dowie on Monday, were without a host of key players against their Midlands rivals, including Cup-tied duo Scott Dann and Daniel Fox.
Under the guidance of Frankie Bunn and John Harbin, the home side led a charmed life in the opening minutes, with both Gera and Bednar coming within inches of giving West Brom the lead.
Gera's downward header in the eighth minute whistled past the post, while a minute later Bednar, just six yards from goal, scuffed his shot wide.
However, West Brom's pressure finally paid dividends when Brunt scored his second goal of the season.
Bizarrely, Coventry appealed for offside from a throw-in and the midfielder imposed maximum punishment by stooping to head Bednar's bye-line cross beyond Marshall.
West Brom's five-man midfield were frustrating the home side; although Coventry did have two chances to level through Michael Doyle and Leon Best.
But the Sky Blues were lucky to go into the break just a goal behind - thanks to the crossbar as Michael Hughes came agonisingly close to deflecting Brunt's driving free-kick into his own net.
Any hopes of a Coventry comeback were effectively ruined when referee Mark Halsey sent off Doyle when, at most, a yellow card would have sufficed.
Gera laid prostrate, clutching his knee, for a good few minutes but replays later showed that the Republic of Ireland midfielder's late challenge had made only faint contact with the Hungary striker.
Marshall gifted a second goal to the visitors when the goalkeeper kicked a pass-back straight to Bednar, who rounded the red-faced Marshall for an easy finish.
And just when Coventry thought things could not get any worse De Zeeuw conceded a penalty when he tried to stop Bednar's surging run. Marshall had no chance of saving the striker's low, well-struck shot.
Substitute Miller showed why the Baggies had paid Manchester City £900,000 for his services in January with a great finish into the corner for West Brom's fourth.
And then Gera, whose every touch had been greeted by a cacophony of boos by the home fans after Doyle's sending off, ended the rout.
Miller was again at the heart of the attack with his square pass finding the Hungarian, who danced past a phalanx of defenders before easily beating Marshall 13 minutes before the final whistle.
Coventry caretaker manager Frankie Bunn:
"We'd clawed our way back into the game and that sending off took it away from us completely.
"Michael said he hadn't done anything untoward and if replays confirm that then we would obviously appeal.
"The red card did kill us but we also made some basic errors."
West Brom manager Tony Mowbray:
"I've no preference who we get in the next round.
"What we need to do now is concentrate on the next game - which is against Hull in the league.
"And that is the priority still because we can't let the possibility of getting to the semi-final of the FA Cup ruin our chances of promotion."
Coventry: Marshall, Osbourne, Ward, De Zeeuw (Birchall 75), Hall, Simpson (Thornton 75), Michael Hughes, Doyle, Tabb, Mifsud, Best. Subs Not Used: Konstantopoulos, Gray, Borrowdale.
Sent Off: Doyle (48).
West Brom: Kiely, Hoefkens, Barnett, Albrechtsen, Robinson, Filipe Teixeira (Kim 74), Morrison, Pele, Brunt, Gera (Beattie 79), Bednar (Miller 70). Subs Not Used: Danek, Phillips.
Goals: Brunt 12, Bednar 59, 69 pen, Miller 76, Gera 78.
Ref: Mark Halsey (Lancashire).
BBC Sport Player Rater man of the match: West Brom's Roman Bednar 7.02 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.
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