Trailing 58-57, Anosike grabbed an offensive rebound and was grabbed by Kia Vaughn from behind.
Unclear whether the foul had come before time expired, the Rutgers bench began to celebrate, but officials reviewed the play and determined that the foul had been committed just before the buzzer.
Television replays showed the game clock seemed to pause as Anosike came down with the ball and two-tenths remained on the clock, leading to the controversial finish.
Anosike calmly stepped up and hit the two free throws to seal the victory.
Candace Parker, who bruised her knee in the Lady Vols' last game, had 27 points and 10 rebounds, and Angie Bjorklund added 13.
Epiphanny Prince had 21 points for Rutgers, Essence Carson added 18.
The Lady Vols (22-1) entered the second half with a 34-23 lead after the Scarlet Knights (19-4) made only two field goals and two free throws in the 11 minutes before the half.
But Tennessee went cold after the break, making only one shot from the field -- a 3-pointer from Angie Bjorklund -- in the first 141/2 minutes.
Rutgers took a 39-38 lead on a Rashidat Junaid layup with 10:08 to go. Vaughn's putback basket put the Scarlet Knights up 56-51 with 1:35.
Coming off a timeout, Shannon Bobbitt nailed a 3-pointer for Tennessee with 1:23 left.
After Matee Ajavon missed a layup on the other end, Alexis Hornbuckle grabbed the rebound, setting up another 3-pointer by Bjorklund to put the Lady Vols up by 1.
Carson responded immediately with jumper with 26 seconds left. Bobbitt and Parker both missed shots on the goal before Anosike was fouled trying to grab the rebound.
Rutgers was coming off an 73-71 upset of previous No. 1 Connecticut and was trying to become the first team to ever beat No. 1 teams in consecutive games.