Mallya was backed by the 27-member JD-S legislature group. The Bharaitya Janata Party (BJP) was also expected to support the businessman but there were some differences in the party over the matter.
Meanwhile, the Bangalore-based businessman B G Uday conceded defeat in Bihar, where polls are being held for five seats.
"I have conceded defeat. I can't survive in politics here. I will go back to Bangalore after congratulating the winners," Uday told media personnel.
Polls for 18 berths in the Rajya Sabha are underway in five states.