Pittsburgh added depth to their rotation on by acquiring Happ from Seattle for minor league pitcher Adrian Sampson.
A little while later, they put struggling starter A.J. Burnett on the 15-day disabled list with inflammation in his pitching elbow.
They got the 32-year-old Happ, who went 4-6 with a 4.64 ERA in 20 starts and one relief appearance for Seattle this season. They then made the move with Burnett, who had his third straight subpar performance during a 15-5 loss to the Cincinnati Reds.
"We talked about all sizes and shapes of deals and decided Happ was our best move," Pirates GM Neal Huntington said.
The 38-year-old Burnett gave up 10 hits, two walks and eight runs -- seven earned -- in 4 1/3 innings in the series opener. The Reds also stole four bases off Burnett, whose velocity was down at the start of the game.
Left-hander Bobby LaFromboise was called up from Triple-A Indianapolis to take his roster spot. LaFromboise made his one appearance with the Pirates this season, pitching a scoreless inning on May 14 at Philadelphia.
Soria arrived in Cincinnati shortly before the 4 p.m. Hurdle plans to use him in a set-up role for closer Mark Melancon, who has converted a club-record 30 consecutive save chances and is 32 of 33 overall this season.
Morse, 33, is making $7 million this year and is under contract for $8 million in 2016. He bats right-handed, making him a complement for Pedro Alvarez. Morse was hitting .213 with a .276 on-base percentage, .313 slugging percentage and four home runs this season for the Miami Marlins. The Marlins had sent him to the Los Angeles Dodgers on Thursday as part of a three-team, 13-player trade."(Morse) is a right-handed bat off the bench who can spell Pedro at first base," Huntington said. "This gives Jose an opportunity with another club. Sometimes, moving on is the best thing for both sides."