The hosts went ahead after 22 minutes when Rick ten Voorde found the net from Thomas Bertels' left-wing centre.
However, Weimann levelled on the hour when he collected a Samir Carruthers' through ball before rounding the goalkeeper Lukas Kruse and slotting home.
Villa manager Paul Lambert was pleased with the performances of Weimann and Carruthers as well as youngsters Josh Webb and Graham Burke.
Webb and Burke featured against the Budesliga 2 outfit after being drafted in as cover for Aleksandar Tonev, Gabby Agbonlahor and Jores Okore.
Lambert told Villa's official website: "It was a great goal from Andi. It was a great ball through from Samir. Andi then did great to round the keeper. It was really pleasing.
"There are a lot of positive things there. We had young Josh coming over with Graham - and Samir is here too.
"You blood one or two young ones. It's good for them.
"Gabby had a slight knock along with Alex and Jores. We have missed those three. But the young lads came in and coped well.
"Winning these games is immaterial. You try to get everyone fit for that first game.
"The main thing for me is getting everyone fit. It's another 45 minutes under the belt for the lads. That's always pleasing."