The right-handed Bailey, the 2009 American League Rookie of the Year with Oakland, pitched a scoreless seventh, allowing one batter to reach on a wild pitch on a strikeout and walking another.
"It felt good," said Bailey. "It was nice to get back out into that environment and get the first one out of the way, for sure."
Bailey, who was 6-3 with a 1.84 ERA and 26 saves in his rookie season with Oakland, was signed by Angels on Aug. 12 as a Minor League free agent. He went 2-1 with a 2.00 ERA in eight relief appearances at Salt Lake.
After missing all of 2014 following shoulder surgery, Bailey spent 2015 in the Yankees organization, including 10 Major-League relief appearances after being called up Sept. 1. With the Phillies this season, he was 3-1 with a 6.40 ERA in 33 relief appearances before being released Aug. 6.
"I thought I threw the ball well," Bailey said. "I got behind some guys, was able to work out of it. That's my biggest thing, just getting back to commanding the zone. But today I was a little amped up, obviously, getting back here. But like I said, it's nice to get that one out."
Ege, claimed off waivers from Miami on Aug. 12, retired all three hitters he faced in the sixth.
The left-hander had a 4.91 ERA in six relief appearances with Salt Lake without a decision, striking out 11 and walking two in 7 1/3 innings.
"Ege looked like he had good angle, he threw some good sliders," manager Mike Scioscia said. "Got threw his inning clean. Good to see those two guys out there and we'll see where they work in our pen."
With the trade of reliever Fernando Salas earlier this week and Huston Street and Cam Bedrosian on the disabled list, the Angels had no one on the active roster with more than one Major League save this season.
"We're going to do it by committee even more so than we have been," Scioscia said before Friday's game. "I think we have some good arms down there. Hopefully, going to get it done and hold leads."
Catcher Juan Graterol also was recalled from Salt Lake. He made his Major League debut in the eighth inning Friday and then had an RBI double in the ninth in his first at-bat.
Graterol was with the Angels from July 18-23, but did not appear in a game. He hit .300 with two homers, 10 doubles and 23 RBIs in 68 games with Salt Lake. He has played in 527 minor league games over 11 seasons without appearing in a big league game.
"Anytime you see a kid come up and get his first hit, it's exciting," Scioscia said. "Juan has improved so much and he's worked so hard to get here. The guys in the dugout felt great for him.