It was part of a busy day for the Marlins in which they made roster moves regarding a total of eight players, including left fielder Christian Yelich, who was placed on the 15-day disabled list.
Heaney, the team's top prospect, will make his big league debut Thursday against the New York Mets. The 23-year-old was the ninth overall pick in the 2012 draft and had a combined record of 7-2 in the minors this season with ERAs of 2.35 in Double-A and 2.74 in Triple-A.
He will try to shore up a rotation that was rocked last month by the loss of ace right-hander Jose Fernandez to season-ending elbow surgery.
"It's surreal," Heaney said. "You think of how it might happen, and then it does, and it doesn't hit you until now when I'm in the park and looking out."
DeSclafani, 24, who made his big league debut earlier this season and went 1-1 with a 5.56 ERA in two starts, will start again Tuesday against the Chicago Cubs.
To make roster room for their two new starters, the Marlins designated for assignment left-hander Randy Wolf, 37, and right-hander Kevin Slowey, 30. Wolf, trying to make a comeback from a second Tommy John surgery, went 1-3 with a 5.28 ERA. Slowey, with 17 appearances, including 15 in relief, was 1-1 with a 5.30 ERA.
Right-hander Jacob Turner, who had been set to start Tuesday, now will pitch out of the bullpen.
Yelich, who had missed the past two games due to a strained lower back, was replaced by outfielder Jake Marisnick, who also was called up from New Orleans. Marisnick will start Monday night in center field for the Marlins, who will move Marcell Ozuna, at least temporarily, to left field.
"Jake's natural position is center," Marlins manager Mike Redmond said. "I feel confident he can play anywhere. But I just felt it was easier for Jake as a young guy to come up and play your natural position. He covers a ton of ground.
"I talked to (Ozuna) about it, and he's fine. Wherever he plays, he plays."
Redmond said he did not think Yelich would require longer than 15 days on the disabled list, retroactive to Saturday.
The Marlins have a glut of utility infielders with Ed Lucas and Jeff Baker already on the roster, and that made Solano expendable. Bour, meanwhile, gives the team a left-handed hitter off the bench, which is something Miami did not have before the move.
Also on Monday, the Marlins announced the signing of the second overall pick in the 2014 draft, right-hander Tyler Kolek.
"We're trying to win," said Redmond, whose team started the day one game out of first place in the National League East. "We felt like bringing the young pitchers in ... this was the time. Hopefully, they will help us win games."