Chicago sent minor league left-handed pitcher Tyler Thomas to the Rangers in return for Chavez.
Chavez, 34, had one save and a 3.51 ERA (22 ER/56.1 IP) in 30 relief appearances with Texas this season. Thomas has gone 3-5 with one save and a 2.88 ERA in 15 appearances, all but one as a starter, for Class A South Bend.
"The word that comes to mind is versatility. You can start him or put him in the bullpen and he's very good in both arenas," Cubs manager Joe Maddon said of Chavez.
When asked if the team was done making moves before the trade deadline, Maddon said: "I don't think it's going to be the end of it. They (the front office) enjoy it too much."
Morrow's move to the disabled list is retroactive to Monday. Chicago kicked off MLB's second half of the season with a 9-6 win over the St. Louis Cardinals on Thursday night.
"It's nothing that can't clear up with rest," Morrow said. "I seem to rotate injuries. I never hurt the same place twice. This is a new one, but it's manageable."
Morrow spent time on the disabled list a month ago when he suffered back spasms while undressing. The right-hander has battled injuries throughout his career. His velocity was down in his last outing on Sunday.
"It was bothering him a bit," Maddon said. "We thought it was manageable but now it's not, so we have to take a little bit of a break. We don't anticipate him being gone for a long time, but it seems to be prudent to go this course right now."
Anthony Bass was activated to take his place on the roster.
Morrow was signed in the offseason after he played one season with the Los Angeles Dodgers and dominated the Cubs in the postseason.
He has appeared in 35 games and has a 1.47 ERA and a career-high 22 saves. Maddon said Steve Cishek, Pedro Strop and Carl Edwards Jr. each could be looked at to close depending on matchups. Strop closed out Thursday's win after Edwards scuffled in the ninth.
Information from ESPN's Jesse Rogers was used in this report.