Vanessa Sahinovic sustained multiple fractures and a head injury - but is not in a life-threatening condition.
Luna Pajer, 15, gained injuries to her arms and has also returned to Vienna for specialist treatment.
The Austrian Olympic Committee (AOC) said the team would still compete.
"It's a very, very sad day," said Dr Karl Stoss, AOC president. "It's a tough day for the team. Sporting motives and goals are not important at this moment."
Event organisers said the matter was now being investigated by police.
Team doctor Dr Robert Kandelhard accompanied Sahinovic and Pajer back to Vienna. The private jet belonging to Azerbaijan president Ilham Aliyev was used to fly the injured athletes back to their homeland.
"They have already landed and can get the best care there," added Stoss. "I'm not a medical expert but the treatment has been really good so far.
"It's a real shock for us and tough to motivate the team to focus on sport."
The AOC said the collision happened at 08:30 local time on Thursday, when the athletes were walking on the pavement in the Olympic village.
Verena Breit, 15, was also injured in the incident and she spent an hour in hospital in Baku with a thigh injury, but has now returned to the athletes' village.
A Baku 2015 spokesperson said: "This is a terrible accident, and at this time our thoughts and concerns are for the athletes involved, their families and the remainder of the Austrian delegation.
"The incident is now a matter of a police investigation and until that process is complete we will be making no further statement."
The inaugural European Games feature 20 sports, 16 of which will be included in next year's Olympics, and begin on 12 June.
Meanwhile, British newspaper the Guardian claims it has been banned from entering Azerbaijan to cover the event.
It says the decision "appears to be linked" to its report on the country's preparations for the event, which contained criticism of the government's "clampdown on freedom of speech and any political opposition".
A 15-year-old Austrian synchronised swimmer suffered "severe injuries" when she and two other teenagers were in a collision with a bus in the athletes' village at the European Games in Baku.