"Rich Rod was one of the few coaches that gave me a chance to play as a quarterback on the next level," Robinson said in an interview posted on mgoblue.com, the school's website, on Monday.
Robinson decided to stay after talking to his parents, brother and high school coach.
"This is my family -- my home now," he said.
Rodriguez was fired Jan. 5 and Brady Hoke was hired a week later. Robinson told Hoke a day after he was hired that he was committing to staying to play for him next season.
Robinson's high school coach said "every school" wanted Robinson to leave and he had heard from 10 to 15 coaches within a day of Hoke's hiring.
The first player in NCAA history to throw and run for 1,500 yards said new coach Brady Hoke was "terrific" and added that he was eager to learn his offense.
"It's going to be something to learn," Robinson said. "It's going to be fine."
Art Taylor, who coached Robinson at Deerfield Beach (Fla.) High School, backs Robinson's decision.
"I'm supportive of Denard and happy for him," Taylor wrote in a text message. "As long as he is happy, that is the most important thing."
Robinson finished sixth in voting for the Heisman Trophy and won awards as the Big Ten's offensive player of the year and MVP.
Michigan athletic director Dave Brandon has said QB Tate Forcier, who backed up Robinson after starting ahead of him when they were both freshmen in 2009, is no longer with the program.