Dadashev's fight against Puerto Rico's Subriel Matias was stopped by cornerman James "Buddy" McGirt at the end of the 11th round on Friday night.
Dadashev was unable to walk to the dressing room after the bout and was immediately hospitalised.
Doctors operated to relieve pressure from swelling on his brain, ESPN and other outlets reported.
"Right now, he's in critical condition, but the doctor told me that he's stable," Dadashev's strength and conditioning coach, Donatas Janusevicius, told ESPN.
"We know that the bleeding has stopped. Also, the swelling has stopped. So some positive news, and we hope and pray for the best."
Dadashev, 28, took an unbeaten 13-0 record into the 140-pound non-title fight.
McGirt said he "couldn't convince" his fighter to stop, but opted to throw in the towel when he saw him "getting hit with more and more clean shots as the fight went on".
"One punch can change a whole guy's life and I wasn't going to let that happen," McGirt said.