The Nigerian stooped to head in Craig Gardner's inviting first-half cross as Tony Pulis' men record their fourth straight win to move 11 points clear of the drop zone
Ideye headed the only goal of the game in the first half to end a three-match winning run in the league for Mark Hughes' men, who were second best before the break, but improved after half-time.
A flurry of injuries - to Ben Foster, Victor Moses and Jonathan Walters - disrupted the second half and Pulis' old club failed to sufficiently trouble replacement West Brom goalkeeper Boaz Myhill.
The hosts could have extended the margin of victory, but wasted countless chances before hanging on to recover from two defeats in the space of four days to Midlands rivals Aston Villa.
Pulis made one change to the West Brom side beaten at Villa in their previous league match, as Stephane Sessegnon replaced Claudio Yacob.
Hughes also made a solitary switch, restoring captain Ryan Shawcross to his defence in place of Philipp Wollscheid.
That backline was soon threatened by Saido Berahino, who fired in two early shots from the edge of the box, which both forced good saves from Asmir Begovic.
The visitors responded with better possession, but no real penetration, and West Brom took a deserved lead in the 20th minute.
Craig Gardner was given space to cross from the right and took full advantage, before Ideye evaded the marking of Phil Bardsley to steer a stooping header into the far corner.
Berahino should have doubled the hosts' tally 10 minutes later.
The striker latched onto a wonderful throughball from Craig Dawson to get in behind Marc Wilson, but snatched at the opportunity, and lashed across goal from a tight angle.
Hughes recognised his side's defensive struggles in the first half and introduced Geoff Cameron at half-time in place of Erik Pieters, moving Wilson to left-back.
That sparked a positive start to the second half, but Stoke's momentum was punctured temporarily as first Foster, and then Moses, required lengthy treatment before having to be withdrawn.
Foster's replacement, Myhill, was thrust straight into the action as he had to backpedal and turn Walters' flicked header over the bar in the 62nd minute - the visitors' first shot on target.
And Begovic was called upon moments later to parry Ideye's low shot at the end of a flowing counter-attack.
Walters was soon added to the injury list, replaced by Mame Biram Diouf in the 68th minute, having taken a heavy blow to the face in an aerial challenge with Dawson.
James Morrison almost added a second for West Brom 10 minutes from time, but Begovic was equal to his powerful effort from 20 yards.
Stoke threw everything forward in the closing stages, and there was a nervous moment for Myhill as he felled Peter Crouch on the edge of the box, but it was not enough and West Brom clung on to beat Stoke for just the second time in their 12 Premier League meetings.