The Braves also sent catcher Anthony Recker and cash considerations to the Twins as part of the trade.
Twins manager Paul Molitor said he planned to start Garcia on Friday night at Oakland.
"We all know his track record. He certainly knows how to pitch. It's a nice addition to our club," Molitor said.
Garcia stopped by the visiting clubhouse Monday afternoon to collect belongings, get a travel itinerary and say goodbye to Braves teammates, who were getting set to face Arizona at Chase Field. He's expected to join the Twins at Dodger Stadium on Tuesday.
Garcia's name had been circulating in trade rumors the past week, and Braves manager Brian Snitker said he handled all of the talk well.
"Seeing his name out there and the game he pitched the other day [seven innings at Los Angeles, just the professionalism he had through the whole thing," Snitker said. "Talked to him today, and he's so appreciative of everything here, as we are of him. He's a great teammate and he did a great job for us."
The Twins have been one of baseball's biggest surprises through the first four months of the season, hovering near the top of the AL Central a year after posting the worst record in the big leagues.
In his last start for the Braves, Garcia gave up three runs and seven hits over seven innings as Atlanta rolled to a 12-3 win over the Los Angeles Dodgers on Friday. Garcia also hit a grand slam, the first for a Braves pitcher since 1966.
Garcia had been scheduled to start Wednesday's series finale at Arizona before the trade.
"He was excited to come here [to Atlanta]," Snitker said, "and he's sad to go. He's a pro, and he'll be an asset to the Minnesota Twins."
García, 31, is 4-7 with a 4.30 ERA, 41 walks and 85 strikeouts in 18 starts for the Braves, who acquired him in a trade with the St. Louis Cardinals last December.
Despite the better-than-expected start to the season, new Twins bosses Derek Falvey and Thad Levine have said they will be active as the trade deadline approaches, but didn't anticipate sacrificing some of their top-rated prospects to land a major name.
Staying true to that public stance, the Twins did not pay a hefty price for Garcia, who is in the final year of his contract.
Ynoa, a 19-year-old from the Dominican Republic, is 0-1 with a 5.26 ERA in six starts for Elizabethton (Tennessee), the Twins' rookie league team. He is 5-11 with a 3.38 ERA in three seasons in the Twins' minor league system.
The 33-year-old Recker has appeared in 206 major league games with four teams, including the past two seasons in Atlanta.
The Associated Press contributed to this report.