Grimm, a rookie right-hander, was 7-7 with a 6.37 ERA in 17 starts this season. His last start was July 12 at Detroit, when he allowed 10 hits and seven runs over 3 1-3 innings before leaving with soreness in his right forearm.
In addition to the former Virginia High ace, Texas sent the Cubs top prospect third baseman Mike Olt and Class-A pitcher C.J. Edwards. There will also be at least one player to be named later, possibly two, depending on who is chosen by Chicago.
The Rangers had long coveted Garza and finally got him a little more than a week before the July 31 non-waiver trading deadline and several days after it initially appeared the teams had a deal in place.
"He's an extremely talented pitcher that's had success in the toughest of divisions and the biggest of stages, who's throwing the ball as well as anybody right now," Rangers general manager Jon Daniels said. "It's a power repertoire, a little bit of a different look from what we've got, and he was available. ... He was in our opinion the best guy on the market."
The 29-year-old right-hander provides a boost to a starting rotation plagued by injuries and a team that slipped three games behind Oakland in the AL West after being swept in a three-game weekend series at home against Baltimore.
Garza (6-1, 3.17 ERA) has made 11 starts since missing the first seven weeks this season with a strained left lat muscle. He is 5-0 with a 1.24 ERA his last six starts.
"We paid a steep price in talent to acquire Matt, but we did so knowing that we've got other guys in the system," Daniels said. "What it comes down to, ultimately why we got him, beside the desire, is we had the players to do it."
Chicago went into Monday night's game 16 games behind St. Louis in the NL Central.