Branyan's grand slam off Rangers left-handed specialist Arthur Rhodes in the sixth completed Arizona's rally from a seven-run deficit.
Branyan added the go-ahead single in the ninth off Alexi Ogando.
What it means for Branyan's future remains to be seen. The veteran first baseman can opt out of his contract on Friday if he hasn't been assured a spot on the Diamondbacks' opening 25-man roster.
Arizona starter Daniel Hudson allowed seven runs -- one earned-- and eight hits with two walks in 2 2-3 innings. His error on a sacrifice bunt attempt helped prolong what turned into a seven-run third inning.