The striker scored two as he continued his remarkable return to form under head coach Tony Pulis.
James Morrison and Saido Berahino also found the net while former Albion loanee Morgan Amalfitano was sent off for the Hammers just ten minutes after coming on.
Ideye, a £10 million record buy, was set to leave the Baggies on deadline day for Qatari club Al Gharafa, but stayed because they failed to land Carlton Cole from the Hammers.
Cole stayed at Upton Park after their move for Tottenham’s Emmanuel Adebayor broke down and since then Ideye has scored four goals in three games.
After an even opening Ryan Cresswell fired a free-kick over but the Baggies took control and deservedly opened the scoring on 20 minutes.
Craig Dawson won Craig Gardner’s crossfield ball ahead of Cresswell to break down the right and, with Ideye free in the centre, found the striker who converted from five yards.
The Nigeria international has admitted he feels reborn under Pulis and he had the Hammers in trouble again soon after, going down under Kevin Nolan’s challenge in the area with replays suggesting it was a penalty.
Albion were dominant and Stephane Sessegnon blazed over, but Gardner almost scored in spectacular style eight minutes before the break. The midfielder collected the ball 35 yards from goal and unleashed a fierce drive which crashed off Adrian’s bar. Gareth McAuley lashed the rebound over, but the Baggies did not have long to wait before doubling their lead.
They had been on top since scoring and grabbed a deserved second when Morrison rode Mark Noble’s challenge and lashed into the corner from 25 yards. A third then arrived on 57 minutes from Ideye when he nodded in Sessegnon’s cross from six yards.
The Hammers had offered little, but Ben Foster had to save well from Diafra Sakho’s header, with McAuley then clearing off the line on 65 minutes. But West Ham imploded with 20 minutes left when Amalfitano – on loan at West Brom last season – was sent off just ten minutes after replacing Nolan. He was initially booked for a foul on Chris Brunt but reacted by pushing the midfielder in the face.
Then, barely two minutes later, Berahino rifled in from a tight angle to spark a mass exodus of the travelling West Ham fans. They had jeered Sam Allardyce’s substitutions and also briefly abused the manager in the second half.
Allardyce brushed off fans’ fury. “Fans get disappointed because there’s 6,500 who have travelled up here and we have built the FA Cup to be one of our big successes this year,” he said. “When you have travelled this way and the performance has not been as good as they have experienced for most of this season they express their disappointment.
“I am going to defend the players because we couldn’t cope with the injury list and fixture list and it has not given us the level we can play at.
“I can’t afford to get disappointed in things like that, it’s about me supporting the players. They have given everything they could. They were dead on their feet.”