But he leaves with 11 Premier League games to go and with United still chasing Europa League glory.
And Rodriguez has revealed the 32-year-old still had goals to achieve in Manchester, but was eventually convinced to head to the States.
Bastian Schweinsteiger has agreed to join Chicago Fire from Manchester United
Bastian Schweinsteiger will be in action from the start of next month
"How was the process? Smooth,” Rodriguez said.
"I think this is a reflection, again, of him. I think it's a reflection of his representation.
"I think it's a representation of how Manchester United conducted itself and allowed us to proceed in these discussions.
"I think it's a reflection of ownership supporting what we are trying to build.
Man Utd fans reacts as Bastian Schweinsteiger signs for Chicago Fire
Tue, March 21, 2017
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"This was smooth. It was long but part of that was down to to Bastian's desire and insistence to remain with United as long as he could.
"He had goals there and he wanted to meet more of those goals.
"He'd just won a championship [the FA Cup] last year with the club.
"But over a time, I think we were able to convince him that our challenge was one that was uniquely suited to his skills and his experience and we're grateful he chose us."
Schweinsteiger is still waiting for clearance to move to America but is hopeful of being in action at the start of April.