The Chilean third seed proved too good for qualifier Mager in the decider, winning 7-6 (7-3) 7-5 after one hour, 34 minutes on the clay in Brazil.
The title win was the fourth of Garin's career and second this month, adding to his Cordoba Open success.
But it was his first at ATP 500 level, having earlier beaten fifth seed Borna Coric in the semi-finals.
Mager, who needed almost three hours to win his semi over Attila Balazs, recovered from a break down in the first set before being edged in a tie-break.
The Italian started the second set well, going a break up and appearing set to force a decider.
But Garin came from 5-3 down, winning 16 of the last 20 points and four straight games to clinch the title.
Down a set when forced off on Friday, top seed Thiem failed to stave off a quarter-final loss once play resumed on Saturday.
Italian qualifier Mager turned his overnight advantage into a 7-6 (7-4) 7-5 victory before again taking to the court later in the day for his semi-final.
The 25-year-old's opponent, lucky loser Attila Balazs, earlier beat Pedro Martinez 2-6 6-4 6-2 but was left clinging onto a place in the tournament when the rain returned, the Hungarian trailing 7-6 (7-4) 3-3 against Mager.
The winner will likely face Cristian Garin in the final after the Chilean third seed opened up a 6-4 4-4 lead over Borna Coric in the other postponed semi-final.
Poor conditions also interrupted the Delray Beach Open, where Milos Raonic was unable to get underway against Reilly Opelka.
The big servers had their semi-final rescheduled for Sunday and will compete for a spot in the decider against Yoshihito Nishioka, who battled from a set down to defeat Ugo Humbert 1-6 6-4 6-0.
Nishioka has reached the final despite dropping the opener in three of his four matches at the ATP 250 event.
"After the first set I just figured out how to play against him and just kept trying to do it and then it worked," he said, as quoted by the ATP Tour.
"If I lose the first set it's okay, I just figure out how he's going to play, how I have to play against him and then from the second set I say, 'Okay, let's do it this way.'"
