With plenty of transfer activity on both sides and a tough pre-season schedule out of the way, managers Ronald Koeman and Mark Hughes will be aiming to get their campaigns off the best possible start with all three points.
Everton's spending spree is meant to off-set the loss of Romelu Lukaku to Manchester United and Rooney ensured the Belgian striker wasn't missed.
Central defender Ahmed Hegazi headed the only goal at the Hawthorns while another newcomer, Jay Rodriguez, kept Bournemouth goalkeeper Asmir Begovic busy in the first half.
The Everton boss, however, expects the same test from the Potters as whenever they have met before since he arrived in the Premier League. I think that all bodes well for the games we have ahead of us. We have spent money but everyone is spending money and looking for the best players. I don't say impossible but 25 goals is a lot and we need to find other ways.
Koeman said: "We know it's a tough start but it's about 38 games, getting the points".
In stoppage-time, Rooney once again took possession in midfield, quickly moving the ball wide before making a perfectly-time burst forward.
Stoke are a classic mid-table Premier League club.
Rooney last scored a goal for Everton at Goodison on March 13, 2004 when he claimed the victor against Portsmouth.
Rooney had already played one home game - a Europa League qualifier against Ruzomberok - in his second spell as an Evertonian following his close-season move from Manchester United.
"They had changes in the team and they lost (Marko) Arnautovic, they brought some new players in ... but Stoke City is always a battle, always a battle home, away".