Ideye, who almost left the club in the final hours of the transfer window, had scored just two goals in 17 games before netting the equaliser against Burnley last Sunday and backed up his midweek goal at home to Swansea with two more against West Ham.
James Morrison and Saido Berahino grabbed the other goals as West Ham had substitute Morgan Amalfitano sent-off for violent conduct in the second half.
Serious crashes on the M40 and M1 motorways meant many fans struggled to make it in time for kick-off. Despite the empty seats in the away end, where 5,500 visiting fans had purchased tickets, the game kicked off at the scheduled time of 12.45pm and it was West Brom who took the early initiative.
Craig Gardner whistled an early tester just wide of Adrian’s post after some tricky work down the left from Morrison.
The goalkeeper may have wished his suspension for being sent off against Southampton had not been rescinded – meaning he was free to play in this game – as West Brom made their pressure count on 20 minutes.
Gardner instigated the move inside the centre circle with a superb diagonal ball that caught West Ham full-back Aaron Creswell napping. His sloppiness allowed Craig Dawson to race down the right wing and deliver a fine cross into Ideye, who slammed the ball in from six yards.
West Ham continued to sit off the hosts and Gardner rattled the crossbar on 38 minutes with a sublime strike from 30 yards.
The visitors failed to heed that warning though – Morrison was given similar space from long range just three minutes later and found the top corner with a wonderful swerving strike after Mark Noble had squandered possession.
Sam Allardyce’s men started to find space between the lines after the break and looked dangerous through Stewart Downing, but they could not muster a chance worthy enough to get back into the game.
And the game was over as a contest on 57 minutes as Ideye took full advantage of some static West Ham defending to score his fourth goal in three games.
Chris Brunt’s cross ran through to Stephane Sessegnon, who kept the ball alive and found Ideye with a pin-point cross. Ideye found space between Cheikhou Kouyate and James Tomkins and nodded home via a small deflection off James Tomkins.
Amalfitano replaced the ineffectual Kevin Nolan, who was booed off by the visiting supporters, but was only on the pitch for 10 minutes as he saw red for a clash with Brunt.
The West Ham midfielder hacked Brunt down from behind and was about to receive a yellow card before he pushed Brunt in the face right in front of referee Martin Atkinson, who promptly sent him off.
Hammers boss Sam Allardyce blanked Amalfitano as he walked down the tunnel and the manager’s mood darkened further seconds later as Berahino bagged his 16th goal of the season.
The youngster beat the offside trap before firing into the bottom corner from an improbable angle, although Adrian should have done better as he was beaten at his near post.
Ideye came close to notching his first West Brom hat-trick on 79 minutes but was denied from close range by Adrian before substitute Callum McManaman curled an effort just wide of the post with two minutes remaining.
West Brom will find out their quarter-final opponents on Monday night when England manager Roy Hodgson makes the draw, while West Ham are left to concentrate all their efforts on the Premier League.