Joshua, who was making the second defence of his belt, knocked the American down with a short right before the referee waved the fight off.
The 27-year-old now has 18 knockouts from as many fights and will face former heavyweight king Wladimir Klitschko at Wembley Stadium on 29 April.
The Manchester Arena was a 21,000 sell-out and those in attendance also included former heavyweight world champions Shannon Briggs and David Haye, and WBC cruiserweight world champion Tony Bellew.
With Molina content to spoil, Joshua found it difficult to get off his shots in the second but he finally found his range in the third, with a devastating outcome.
But the American only lasted a few more seconds, referee Steve Gray calling a halt to proceedings as Joshua was raining blows upon the challenger.
“It was slow, patient and calculating. You are dealing with someone who is not giving you many options, so I had to create them and, once I saw it, I managed to connect properly and knocked him out,” said Joshua.
“He had maybe two shots, but he is not a Kiltschko, he is not a Haye, he is not all of these other fighters.
“That’s why I don’t get too hyped, that’s why I took him out like I should. I am going to keep on handling business like I should.”