Bivol claimed the win with a unanimous decision on points, with all three judges scoring the bout in his favor while he retained his WBA light heavyweight title.
The Russian extended his unbeaten record to 20-0, while handing Alvarez only the second defeat of his career, and his first in nine years since losing to Floyd Mayweather Jr.
“If you don’t believe in yourself what do you do? You get nothing. I believe and my team believed in me.
“I felt his power. You can see on my arm, he hit my arm but not my head. It’s better.”
He previously fought in the light heavyweight division, defeating Sergey Kovalev in 2019.
Last year, Alvarez became the first undisputed super middleweight champion with an 11th round TKO victory over then-unbeaten IBF title holder Caleb Plant.
He has won titles in four different weight divisions: light heavyweight, super middleweight, super welterweight and middleweight.
But despite a passionate, mostly Mexican crowd cheering him on, the 31-year-old fought Bivol, being beaten and injured by the clinical Russian.
Alvarez ended the unusually red fight after the Bivol barrage, but was magnanimous in the defeat.
After the fight, Alvarez said he was eager for a rematch. When asked, he said, “Of course I do. It doesn’t end like that.”
Bivol, who has held the light heavyweight title since 2016, said he was also eager for a rematch.
“I’m ready for a rematch, I just want to be treated like the champion now.”