The Nigeria international striker - who has found form following the recent appointment of Tony Pulis at The Hawthorns - scored either side of James Morrison's first-half stunner to help seal progression against 10-man West Ham.
The hosts led after 20 minutes courtesy of Ideye's third goal in as many games and again made the most of their first-half dominance to double the advantage before the break.
While Adrian got a touch to Morrison's long-range strike shortly before the interval, the West Ham goalkeeper could not keep it out, with Ideye's second early in the second half effectively ending the game as a contest.
West Ham - without the injured trio of Andy Carroll, James Collins and Winston Reid - almost gained a foothold when Gareth McAuley cleared Diafra Sakho's header off the line.
However, the dismissal of Morgan Amalfitano and Saido Berahino's 72nd-minute strike capped a dismal outing for Sam Allardyce as Pulis' men eased into the last eight.
West Ham had the first opportunity as Mark Noble - recalled to the starting XI by Allardyce - earned an early free-kick that Aaron Cresswell curled over before the home side saw penalty appeals waved away as Kevin Nolan appeared to handle.
Craig Gardner was named as West Brom's replacement for cup-tied midfielder Darren Fletcher and the former Sunderland man fired wide on the volley soon after in an enterprising start for the hosts.
When Enner Valencia was fouled by Morrison, Noble this time fired his free-kick over, before West Brom took the lead.
Craig Dawson crossed for Ideye in the centre and, amid suspicions of offside, the in-form forward tucked home the opener to pave the way for West Brom's first prolonged spell of pressure.
Morrison forced Adrian into a smart save before Gardner came close to making it 2-0 when he crashed a long-range strike against the crossbar eight minutes before the break.
However, West Brom's positivity in midfield paid off when Morrison surged to the edge of the area and unleashed a swerving strike that gave Adrian no chance.
Ideye needed just 12 second-half minutes to double his tally - making the most of Stephane Sessegnon's deflected cross to head past Adrian and put the game beyond the visitors.
Sakho saw a header saved by Ben Foster, before McAuley cleared off the line, with Allardyce's misery compounded when substitute Amalfitano was shown two yellow cards.
The midfielder replaced Nolan, but, after being booked for a swipe at Chris Brunt, he was reprimanded again for pushing the West Brom captain in the face as the two clashed in the flashpoint moment.
Berahino added a fourth against the London club - who are now winless in four matches - as Pulis' side remain on course for a repeat of their 2008 semi-final run.
