Texas also received cash considerations in the trade.
The Blue Jays had designated Grilli, 40, for assignment on Tuesday after he had a 6.97 ERA in 26 appearances this season. He was 2-4 with one save.
Pinto, 22, was hitting .311 with four home runs and 18 RBIs for Down East, the Rangers' high Class A team in the Carolina League. The left-handed hitter is in his sixth professional season after being signed as an undrafted free agent.
The Rangers are in desperate need for bullpen help as they lead Major League Baseball with 17 blown saves this season.
Grilli is expected to join the Rangers on Monday when they return to Texas from Chicago, where they were facing the White Sox on Sunday.
The Rangers said they would make a corresponding move Monday to clear room on the 25-man roster. They designated right-hander Preston Claiborne, who is on option at Triple-A Round Rock, to make room on the 40-man roster for Grilli.
Although he has struggled this season, Grilli had a 3.64 ERA in 46 appearances last season for the Blue Jays and has a 4.18 ERA in 15 seasons. He was an All-Star in 2013 when he had 33 saves for the Pittsburgh Pirates.
The Associated Press contributed to this report.