The 26-year-old centre-back was hit by flying glass after three explosions shattered windows on the Dortmund team bus.
"We have learnt that everything went well," Dortmund's president Reinhard Rauball told broadcaster NTV.
The 26-year-old centre-back was hit by flying glass after three explosions shattered windows on the Dortmund team bus en route to Signal Iduna Park for Tuesday's Champions League quarter-final, first-leg.
The match has been postponed until Wednesday at 1845 (1645 GMT), which Bartra will miss, and German daily Bild claim he will be sidelined for the rest of the season.
The player's father -- Jose Bartra -- says he has spoken to his son, who expects to leave hospital in a few days.
"We spoke yesterday after the operation and he told us he was fine," Jose Bartra told Spanish media.
"The doctors had been through and said it was a clean break, that he didn't have anything in his arm and that he would be in the hospital for a few days and then he would be released to go home."
His father said Bartra is shocked by the attack.
"It's true it was a shock… the first thing he heard was a loud bang and an explosion," said Jose Bartra.
"Then he began to feel a pain in his head and his arm hurt a lot because he didn't know what had happened."
German investigators are hunting for possible suspects responsible after Dortmund city's police chief described it as a "targeted attack" against the team.
Investigations will focus on a letter claiming responsibility for the attack that was found close to the site of the blasts.
Dortmund's CEO Hans-Joachim Watzke says the rescheduled match on Wednesday must take place as a sign against terrorism.
"We will play not only for ourselves today. We will play for everyone... we want to show that terror and hate can never determine our actions," said the Dortmund boss.
"And of course we play for Marc Bartra, who wants to see his team win!"