Moussa Dembele's terrific volley gave Lyon the lead on seven minutes at Groupama Stadium, with former Liverpool winger Shaqiri crossing for Jason Denayer to head in a second just past the hour.
Boateng, who left Bayern Munich this summer after 10 years at the club, came on for the final 30 minutes following his arrival on a free transfer.
The 2014 World Cup winner was convicted on Thursday of assaulting his ex-girlfriend during a Caribbean holiday three years ago, and fined 1.8 million euros ($2 million) by a court in Munich.
Lucas Paqueta added a third for Lyon by sweeping in a pass from Houssem Aouar, with Habib Diallo grabbing Strasbourg a late consolation from the penalty spot after a handball.
Lyon, who failed to win any of their first three games, moved up to seventh, seven points adrift of leaders Paris Saint-Germain.
Nice remained unbeaten after a 2-0 win at Nantes courtesy of late goals from Denmark forward Kasper Dolberg and Amine Gouiri.
Nice are fourth but have a game in hand after their August 22 match with Marseille was abandoned in the wake of a pitch invasion by Nice supporters.
The club was docked a point this week following the ugly incidents, and league officials ordered the match to be replayed behind closed doors at a neutral ground.
"We were not happy. We were on the pitch and we feel punished," said Nice captain Dante.
"But in life you can't let your head drop. Especially after an injustice or bad decisions. We have to work hard and do our best this season."
Nice, backed by British billionaire Jim Ratcliffe, will not appeal the ruling. They were leading the Mediterranean derby 1-0 when the game was interrupted in the 75th minute.
"I didn't feel like the group was affected," said Nice coach Christophe Galtier. "But obviously they would have preferred to keep the points they could earn on the pitch."
Highlights Ligue 1 strasbourg Olympique Lyon