The Baggies were made to pay for some glaring misses at Villa Park as the hosts recorded their second win over their Midlands neighbours in the space of five days.
Claudio Yacob and Jack Grealish, for his second bookable offence for diving, were sent off and Villa fans stormed the pitch prior to the final whistle as the officials lost control of a feisty derby in the latter stages.
Saido Berahino, the joint-top scorer in this year's competition, also had an early opportunity to give the visitors the lead as he shot low and hard straight at Shay Given after running away from Ciaran Clark.
Given, again recalled at the expense of Brad Guzan for cup duty, had to be alert to backtrack and tip a deflected Chris Brunt volley over the crossbar moments after he comfortably saved Ideye's header.
Villa were missing injured striker Christian Benteke, who scored the stoppage-time winner in this week's Premier League instalment of the fixture, and struggled to create any openings despite Delph's best efforts.
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It was Delph who provided the Villans with their breakthrough on 51 minutes following great work from Charles N'Zogbia, who cut in from the right and teed up his captain to drill through the legs of Boaz Myhill from 18 yards.
Still it was Albion who were creating the majority of the chances and defender Joleon Lescott somehow headed wide from close range after he had anticipated Clark's inadvertent flick-on to the back post.
Veteran stopper Given, the Villa hero as they overcame Leicester City in the previous round, continued to show his worth and scampered quickly off his line to thwart Berahino at the feet of the 17-goal striker.
The confidence boost from their goal took 10 minutes in coming, but the hosts finally started to play with freedom as Sinclair was denied by Myhill from either side of the box and Ashley Westwood had a shot blocked.
They were given a break on 80 minutes when Yacob collected the second of two quick yellows for a block tackle which left Leandro Bacuna writhing on the floor, and Villa soon capitalised on their man advantage.
Grealish led a counter which saw Sinclair released on goal, and the winger, who spent last season at The Hawthorns, checked back onto his right foot to send a shot inside the far post and his side to Wembley.
Things ended sourly for substitute Grealish when referee Anthony Taylor showed him a second booking for simulation, but that did not stop the Villa fans from celebrating on the pitch either side of the final whistle.