Cotto produced his 31st knockout to move to a 38-4 record with a furiously attacking display.
Fighting under new trainer Freddie Roach, Cotto produced an onslaught of punches in the second round and then the decisive left hook to the head forced the referee to stop the bout 18 seconds into the third.
"It feels good to win and be able to do it in front of a wonderful crowd," Cotto said. "I thought I hit him with some good shots early and I was able to get the stop."
Rodriguez believed the stoppage came too early, but managed to compliment Cotto on his performance.
"The referee really never gave me a chance to get up and show that I could fight back,” Rodriguez said.
"But Miguel showed me his power in the second round and I'm just going to have to regroup."
It was Cotto’s first fight since he was unanimously outpointed by Austin Trout in December while Rodriguez’s record dropped to 28-7-3.