Union had appointed Bjelica as a replacement for Urs Fischer in November last year when the side was reeling in 17th place. They are now 15th after four defeats in five matches with two games remaining.
Grote, who had been a youth coach with the club's Under-19s, had also taken over Union on an interim basis in November.
"I'm looking forward to the task and am convinced that we'll manage to avoid relegation. I know our team well and know that if we join forces we are capable of picking up the necessary points," Grote said in a statement.
Union said assistant coach Marie-Louise Eta, the first female coach at a Bundesliga club, will remain in her position.
Union next visit fellow relegation-threatened side FC Koln on Saturday before concluding their campaign at home against seventh-placed Freiburg on May 18th.
