All three judges scored the fight 115-113 in favor of Bivol.
The loss is just the second of Alvarez's illustrious career and first since he fell to Floyd Mayweather in 2013.
Bivol (20-0, 11 KOs) utilized his jab to take away Alvarez's time and space, frustrating the Mexican throughout the fight.
Canelo landed only 84 total punches, which are the fewest in any 12-round bout in his career.
Overall, Bivol landed 36% of his power punches, while Canelo could muster only 10 total jabs.
Alvarez (57-2-2, 39 KOs) was expected to fight Gennady Golovkin in a highly anticipated trilogy bout in September, provided he defeated Bivol. It's unclear what the 31-year-old's next move will now be, though he said post fight that he will exercise his rematch clause for a return fight with Bivol.