Skaggs (1-0), recalled from Class AAA Reno before the contest, stymied the Rangers on three hits while striking out nine. The 21-year-old left-hander worked around three walks in posting his second career win.
Bell, however, retired Nelson Cruz and Mitch Moreland with two runners on base to end the game, picking up his ninth save.
Cody Ross went 2-for-4 with a solo homer and Paul Goldschmidt also provided two hits and an RBI for National League West-leading Arizona, which took the opener of a home-and-home interleague series.
Rangers starter Martin Perez (0-1), also making his first start of the season, yielded four runs (three earned) on nine hits over 5 1/3 innings.
Adrian Beltre drove in a pair of ninth-inning runs for AL West-leading Texas.