The selection is a Competitive Balance Round A pick that the Marlins basically received for finishing 62-100 in 2013.
To make 40-man roster room for Morris, Jose Fernandez was transferred to the 60-day disabled list. Fernandez is out for the season after undergoing Tommy John surgery in mid-May.
After the Marlins lost, 4-2, to the Braves on Sunday, reliever Arquimedes Caminero was optioned to Triple-A New Orleans. That cleared a spot on the 25-man roster for Morris, who is expected to arrive on Monday in time for the series opener with the Rays.
Marlins president of baseball operations Michael Hill said the trade was to provide immediate help to a club that has remained in contention in the National League East.
"I don't think we're finished in trying to improve our club," Hill said. "We're still trying to make as many improvements as we think we can to help this team. This is the first piece, in bringing Morris aboard."
Morris, 27, is a controllable player in terms of service time for Miami. He has yet to reach arbitration. He opened the season at Triple-A Indianapolis.
The right-hander has appeared in 21 games with the Pirates, posting a 4-0 record with a 3.80 ERA, allowing 10 earned runs in 23 2/3 innings.
In parts of two big league seasons, he is 9-7 with a 3.46 ERA in 81 career games, with 57 strikeouts to 42 walks, and a .250 average against.
"I'm excited," manager Mike Redmond said. "We definitely need another arm down there."
The bullpen took a hit when Carter Capps went down with a right elbow sprain. Capps is getting a second opinion on Wednesday from Dr. James Andrews, and Tommy John surgery is a possibility.
"Losing Capps has been tough," Redmond said. "There's no doubt that we struggle down there in the bullpen. We felt like [Morris] was the guy that will help us. Hopefully he will be able to go out there and log some innings for us.
"I know Pittsburgh has a deep bullpen, so this is probably a huge opportunity for him to get out of a deep bullpen and get to a bullpen where we can really use him."
The Marlins had the flexibility to make the trade because the team had four of the first 43 picks in the Draft. Miami holds the Nos. 2 and 36 picks also.
Only Competitive Balance picks can be traded, and there are 12 total. The slot allotted for the No. 39 pick is $1,457,600.
"For clubs that don't have multiple picks, clubs who just have one pick per round, it was attractive," Hill said. "We tried to take advantage of that, if there was something out there that made sense for us. We had a need. It was in our bullpen. We were looking for the piece that was the right fit for us in terms of controlling, not just short-term, but long-term, and he had the stuff to help our club as well."