Rodriguez battered Edwards throughout the fight at the Desert Diamond Arena in Glendale, Arizona, finally connecting with a massive left to the chin that saw the Briton pitch forward to his knees, his face on the canvas.
He straightened up, but as the referee administered the count Edwards' corner signaled him to call a halt moments before the bell to end round nine.
"I had a lot of motivation going into this fight knowing I have a daughter on the way, I'm just so happy right now," said Rodriguez, who improved to 19-0 with 12 knockouts.
"I'm just satisfied that I got the win, period," he said.
"That was my main goal. But during camp I knew I had what it took to knock him out or stop him and that's what I did, so (that) just made the win that much better."
Edwards, who fell to 20-1 with four wins inside the distance, had cuts around both eyes by the time the action-packed bout was halted.
The cut over his left eye, he said, made it hard to get away from Rodriguez's lead hand.
"I think it was a great fight, but, yeah, the better man won today," said Edwards, who had made four successful defenses of the IBF title he won with a decision over South Africa's Moruti Mthalane in 2021.
Rodriguez had claimed the vacant WBO title in April with a decision over Cristian Gonzalez Hernandez.
He had moved down in weight after claiming the World Boxing Council super flyweight world title with a victory over Carlos Cuadros in February of 2022, a title he defended twice.
