The Phillies fell behind 5-0 in the first three innings as Lee struggled.
Philadelphia rallied by scoring two runs off Bronson Arroyo in the seventh and four more in the eighth against Joe Thatcher and Trevor Cahill (1-5).
Jeff Manship (1-0) allowed a hit and two walks in an inning and Jonathan Papelbon got an assist from a replay in the ninth to earn seventh save.
Cody Ross had three RBIs and two hits for Arizona, which has lost nine of 10 at home.
The Diamondbacks thought they had a runner on second against Papelbon in the ninth when Martin Prado, who had singled, was called safe at second after a pitch was thrown in the dirt. After a review, the umpired ruled Prado came off the bag and was tagged by second baseman Chase Utley.
Papelbon then struck out Paul Goldschmidt looking, and Miguel Montero grounded out to end it.
Off to the worst start in franchise history, the Diamondbacks started to find their groove this week.
Arizona won the final two games of a four-game road series against the Cubs and opened the weekend series against the Phillies with a 5-4 win, ending an eight-game home losing streak.
To keep it rolling, the Diamondbacks would have to knock off Lee, something that hasn't been easy for them in the past.
In six previous starts against Arizona, the left-hander was 4-1 with a 2.93 ERA and 42 strikeouts in 43 innings. Lee has been dialed into the strike zone against everyone so far this season, entering Saturday night's start with 38 strikeouts and two walks.
Lee's control wasn't quite the same against the Diamondbacks.
Arizona scored two runs off Lee in the first, on Goldschmidt's sacrifice fly and an RBI single by Ross that ended a 1-for-22 slump since returning from the DL with a dislocated hip. A.J. Pollock added a run-scoring triple in the second inning and Ross made it 5-0 in the third with a two-run single to left.
Lee allowed three earned runs and eight hits in six innings. He also walked two to double his season total.
Arroyo did his part, allowing two runs and seven hits in 6 1-3 innings, but Arizona's bullpen let it slip away.
Thatcher allowed two runs before being lifted and Cahill labored for the first time since moving to the bullpen. Asche hit a two-run double that hit first base umpire Bob Davidson before caroming toward the stands and Ben Revere followed with a run-scoring single that made it 6-5.
NOTES: Phillies 3B Freddy Galvis is hitless in 21 at-bats after going 0 for 4. ... RHP A.J. Burnett, Philadelphia's starter in Sunday's series finale, has yet to win this season in five starts despite a 2.73 ERA. ... RHP Brandon McCarthy, Arizona's starter on Sunday, tossed eight scoreless innings against the Phillies last season, but did not figure in the decision.
