After Gaudin (2-1) notched a win over St. Louis last week in his first start in three years, the converted reliever again pitched well for the Giants, holding the Diamondbacks to two runs on five hits.
The right-hander walked one and struck out seven as San Francisco took two out of three in the match-up between National League West contenders.
Marco Scutaro went 3-for-4 with a solo homer and Brandon Belt also supplied a bases-empty blast for the Giants.
Diamondbacks starter Tyler Skaggs (1-1) was roughed up for five runs on eight hits in five innings.
A.J. Pollock delivered Arizona's only RBI.