Zagurski, who was out of Minor League options, cleared waivers out of Spring Training despite only giving up three runs in 10 1/3 innings.
He had a productive start to the season in Reno, so when Joe Paterson served up 11 runs in his first 2 2/3 innings of the year, Zagurski got the first crack at filling the position in the D-backs' bullpen.
"Dating back to Spring Training, when I was sent down and I cleared waivers, I was pretty much their property," he said. "But I felt like they called me up pretty quick, which means they considered me first. Once I got called back up, I took it as I was the guy they were looking for. I'm going to do as well as I can, and hopefully it'll work out."
On most occasions, Zagurski has come into games when the D-backs were trailing by multiple runs. In fact, in all but one of his 10 appearances, the team has lost.
"He's filled his role well," Gibson said. "We gave him quite a bit of activity, and his arm holds up well. He gets ready quick, left him in there against righties, he's been good. He fits in the bullpen well."
Zagurski hopes by pitching well when the game is not on the line, he'll be looked at down the road in tighter situations.
"Hopefully I can continue to pitch well and get into some more important roles," he said. "Come into some bigger situations where we need to get an out and be that guy. I hope I'm going in the right direction.
"When I first got here there were five main guys in the bullpen, and I didn't do a great job of taking the load away from those guys," he said. "At the worst case, I have to eat a couple innings here and there to give those guys days off. That's the job I'm at right now, and hopefully it'll evolve into something more important."