Webb was recalled by the Diamondbacks on Wednesday.
Webb was claimed off waivers by the Diamondbacks in mid-April, and he made five relief appearances at Triple-A Reno. He posted a 12.46 ERA and 2.44 WHIP in 4.1 innings but will join the major-league club as bullpen depth after southpaw Kyle Nelson (undisclosed) was placed on the injured list Wednesday.