The hosts dominated the early passages of play and they might have been ahead inside 10 minutes, when debutant Jay Rodriguez broke down the right before picking out Matt Phillips inside the box, only for the frontman to skew over the bar from a tight angle.
Rodriguez was at the heart of all of West Brom's forward movement and he almost notched the opener with a snapshot just inside the area, leaving Asmir Begovic to react quickly in order to turn the ball away from goal with his knee.
The 28-year-old frontman was at it again moments later and this time Rodriguez had the beating of Begovic, although his curling effort went the wrong side of the far-post.
Tony Pulis was full of animation on the sidelines as he encouraged his side to make use of their first-half dominance, which finally paid off when new centre-back Hegazi leapt highest in the box to head a Chris Brunt corner beyond the Bournemouth stopper.
The Cherries appeared rather out of sorts at The Hawthorns, and it took the best part of 40 minutes for the visitors to fashion a noteworthy chance, Ryan Fraser crossing promisingly for Benik Afobe, whose glancing header flew wide.
The Baggies should have doubled their lead just before the break when Hegazi headed a knockdown for Brunt, but the winger somehow managed to push his effort straight across the face of goal without troubling Begovic.
Eddie Howe's side enjoyed some long spells of possession in the second half without making an impact, and Claudio Yacob almost made the away side pay for their wastefulness, as the defender headed agonisingly wide after losing Joshua King in the box.
Sensing that the game might be escaping his men, Howe introduced club-record signing Jermain Defoe from the bench, but he was unable to alter proceedings as the Baggies shut up shop to record a well-earned win on the opening Saturday of the new season.