With the squad en route to Signal Iduna Park ahead of their Champions League quarter-final first leg with Monaco, three explosions damaged the bus and fractured Bartra's wrist and lodged shrapnel in his arm.
Bartra is now not expected to return for "several weeks", likely ruling the former Barcelona man out of the rest of the season, with Dortmund releasing a statement on Wednesday following a successful operation.
The 26-year-old posted an update for fans on social media, thanking them for their support.
"Good news regarding Marc Bartra," the club statement read. "The BVB player, who sustained injuries to his arm and hand in a bomb attack on Tuesday, is recovering after his operation.
"'Given the circumstances, he is doing well,' said sporting director Michael Zorc after paying the Spaniard a visit in hospital.
"Bartra has been absolutely delighted and visibly moved that so many teammates and club officials have paid him a visit since yesterday
"He will be watching tonight's match on television and has his fingers crossed for the team. Following the successful operation, the Spaniard will be now out of action for several weeks as he recovers."
German police are investigating possible extremist links following conflicting letters claimed responsibility.
Fans of Dortmund opened their homes to Monaco's stranded supporters for the night after the first-leg game was postponed for 22 hours.
Using the hashtag #bedforawayfans on social media, many fans from the south of France found emergency accommodation.