Dan Uggla hit a three-run shot, and the Braves snapped Lee's streak of 11 straight games pitching at least seven innings and allowing three earned runs or less.
But Lee (10-2) still won his eighth straight decision. He allowed four runs and eight hits in 6 1-3 innings. J.C. Ramirez and Antonio Bastardo combined for five outs. Jonathan Papelbon finished for his 18th save in 22 tries.
Right fielder John Mayberry Jr. helped Papelbon with a sliding catch on Brian McCann's sinking liner for the second out.
Paul Maholm (9-7) gave up five runs - four earned - and nine hits in six innings. Tim Hudson was scheduled to start for the Braves, but he was pushed back one day because of a stiff neck.
Desperate to avoid falling further behind in the NL East standings, the Phillies (42-45) opened an important 10-game homestand before the All-Star break looking to gain ground. The first seven games are against the two teams above them in the division.
The division-leading Braves have lost three in a row. The Phillies have won three of four, including two of three at major league-leading Pittsburgh.
Lee cruised into the seventh with a four-hitter before running into trouble. Lee hit Freddie Freeman, and Chris Johnson followed with a single. Uggla then hit a drive to left-center to get the Braves within 5-3.
After Lee gave up a one-out double to McCann, pinch-hitter Reed Johnson singled to chase him. Ramirez came in and allowed a sacrifice fly to Andrelton Simmons that cut it to 5-4.
Quintero drove an opposite-field shot out to right to make it 3-0 in the second. A Braves fan dropped the ball back onto the field, and Quintero stopped at second being awarded the homer. The ruling was upheld after the umpires looked at a video review.
Howard connected in the third, hitting a towering shot way out to right-center for his 11th homer this year.
Quintero's RBI double drove in Delmon Young for a 5-0 lead in the sixth.
Braves catcher Gerald Laird was ejected by plate umpire Angel Hernandez after getting called out on a pitch that appeared high and outside in the third. Laird was tossed after shouting an expletive as he walked toward the dugout. He turned around and ran at Hernandez, going nose-to-nose in a heated exchange while manager Fredi Gonzalez tried to calm him. Gonzalez also was ejected.
A throwing error by first baseman Freeman gave the Phillies a 1-0 lead in the first.