The Puerto Rican endured a forgettable 2012, losing his previous two bouts against Austin Trout and Floyd Mayweather Jr on unanimous decisions.
But the 32-year-old produced the 31st knockout of his career to improve his record to 38-4 with a dominant display against Dominican opponent Rodriguez.
Cotto - fighting for the first time with new trainer Freddie Roach in his corner - was on the offensive from the start and landed with a series of left hooks to the body.
He hurt his rival towards the end of the second, producing a strong combination that ended with a left hook to the head that clearly staggered Rodriguez.
The end was not much further in coming, Cotto putting his man down in the next with a flurry of head shots that prompted referee Frank Santore Jr. to step in and wave the contest off just 18 seconds into the third round.
Cotto can now start to think about the possibility of some much bigger dates with a fight against Saul 'Canelo' Alvarez - who lost to Mayweather last month - already being mooted.
The delighted Puerto Rican said afterwards: "All of the work that we put in over the last month really showed tonight."
Rodriguez insisted that the contest had been waved off prematurely, but was quick to praise Cotto for 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."