The roadside blasts left Dortmund's Spanish international Marc Bartra and a policeman injured, with the bombs "containing metal pieces" detonating minutes after the team bus set off to a planned Champions League game against Monaco on Tuesday night.
The blast had a radius of more than 100 metres (yards), federal prosecutors said, adding it was lucky the toll was not more severe.
The match was put back to Wednesday as security was ratcheted up around Dortmund and in Munich where Bayern Munich will take on Real Madrid.
But a defiant Dortmund vowed not to "give in to terror", with players returning to the pitch for training.
Extra forces were deployed around team hotels and their buses will take designated safe routes to the stadiums.
UEFA said "security procedures will be enhanced accordingly wherever needed".
Federal prosecutor's office spokeswoman Frauke Koehler said the probe was examining a possible "terrorist link", after three identical letters were found at the scene.
"An Islamist background appears to be possible," she said, noting the letter demanded that Germany withdraw its deployment of Tornado reconnaissance missions in the anti-IS international coalition and close the US air base in the western German town of Ramstein.
"Two suspects from the Islamist spectrum have become the focus of our investigation. Both of their apartments were searched, and one of the two has been detained," she added.
The assault was described by Dortmund city's police chief as a "targeted attack" against the team, while Chancellor Angela Merkel said she was "horrified" by the "repugnant act".
But Dortmund's chief executive Hans-Joachim Watze vowed that his side "will not give in to terror".
"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," he said.
Monaco's vice president Vadim Vasilyev said "football must not be taken hostage", and pledged that the quarter-final would go ahead.
In a show of solidarity, German Interior Minister Thomas de Maiziere will also attend the match.
Germany has been on high alert since a series of jihadist attacks last year, including the Berlin market assault.
The explosives detonated minutes after the Dortmund team bus pulled away from the squad's hotel and headed for the stadium.
Bartra underwent surgery on a broken wrist after he was hit by flying glass, Dortmund president Reinhard Rauball said.
A policeman, who was on a motorcycle escorting the team bus, suffered trauma from the noise of the blasts.
"We are assuming that they were a targeted attack against the Dortmund team," said the western German city's police chief Gregor Lange.
The explosives shattered the bus windows and the vehicle was burned on one side.
"The bus turned on to the main road, when there was a huge noise – a big explosion," Dortmund's Swiss goalkeeper Roman Burki told Swiss media.
"After the bang, we all crouched down in the bus. We did not know if more would come." Some players hurled themselves to the ground, he said, adding that Bartra was "hit by splinters of broken glass".
The announcement that the game was postponed was only made to the stadium packed with stunned fans including families with young children about 15 minutes before kick-off.
As Monaco supporters cheered Dortmund and wished the players well, some Dortmund fans also took in stranded Monaco supporters for the night.
Bild put out a full-page advert in Dortmund's yellow and its BVB 09 logo, with the message: "You'll never walk alone".
Rauball said he believed the team would be ready for Wednesday's game.
"The worst thing would be if whoever committed this attack was now able to get to affect them through it," he said.
But ex-Dortmund player Steffen Freund, who won the Champions League with Borussia in 1997, warned such a direct attack would not be forgotten by Wednesday.
"Mentally and psychologically that is hard to absorb, it's a lot to deal with."