Villa, who kicked off last season with a 3-1 win at Arsenal, had to defend well in the early stages, but looked increasingly threatening as the match opened up.
And Weimann found the target five minutes into the second half, sweeping home a right-footed effort from inside the penalty area amid some questionable Stoke defending to secure all three points.
Stoke lost just three times at home last season, and Saturday's result represents their first opening-day defeat since 2010.
The Villa defence, which also featured new boy Philippe Senderos for the first time, came under pressure early on, with Steven N'Zonzi sending a tame effort into the arms of goalkeeper Guzan before Bojan drilled wide of the left post.
N'Zonzi and Bardsley both required treatment after being caught by strong challenges during the first half, as new Villa assistant manager Roy Keane – capable of a full-blooded tackle himself during his playing days – watched on from the dugout alongside Lambert.
Bardsley saw a curling effort well saved by Guzan in the closing minutes of an open first half, before Stoke had to scramble the ball clear after Charles N'Zogbia had an effort blocked.
The home side were not so successful in clearing their lines five minutes into the second half, though, as the ball fell kindly to Weimann and he fired across Begovic into the back of the net to stun the home supporters.
Bardsley and Weimann had a coming together in the Villa box on the hour-mark, but the Stoke defender's protestations for a penalty were ignored by referee Anthony Taylor.
