Hegazi, on loan from Egypt's Ahly, scored the only goal of the game in the 31st minute, deftly heading Chris Brunt's inswinging ball from the right between the legs of Bournemouth keeper Asmir Begovic.
"It is a very good win and a fantastic feeling", the 26-year-old player said on the club's official website.
"We know what we're after and we are desperate to bring them in, but must be patient". We have 17 senior professionals and we are at the top age as well, so it just needs restructuring and everybody is aware of it.
Jay Rodriguez, another off-season arrival for the Baggies, had gone close on a couple of occasions prior to Hegazi's header, while Bournemouth looked uncharacteristically bereft of attacking verve in their quest for an equaliser. He has brought a lot of energy to the football club.
"I am disappointed. It is unlike us to have so much possession and to not create chances - that is the most disappointing thing", he said.
The Argentina global went close again early in the second period with a header into the side-netting as the Baggies continued to pose a threat from dead balls.
He then retreated to his job at the heart of West Brom's defence where he got involved in a physical matchup with Bournemouth's Congolese attacker Benik Afobe whom he contained admirably. We gave away a needless free-kick, and against West Brom that is a risky thing. I thought we were ready.
"As for the season ahead, we of course want to improve our Premier League challenge and do everything we can to make a strong challenge in at least one of the major cup competitions".
"I expected us to have the possession that we did but I thought we would have been more creative with it".