Lee (4-4) pitched effectively into the seventh inning for his first home victory of the season.
Devin Mesoraco had a career-high four hits for Cincinnati, including two doubles and an RBI. The Reds again played without slugger Joey Votto, who remained in Cincinnati during the three-game series with a left knee injury.
Philadelphia's offense busted loose for a second straight day. Despite playing in hitter-friendly Citizens Bank Park, the Phillies entered Sunday second-to-last in the NL in home runs with 27.
But they were riding the confidence of homering twice while scoring a dozen runs in Saturday's victory, which snapped streaks of four straight losses and 23 consecutive scoreless innings.
"Today's game was a momentum game and a build off (of Saturday)," Philadelphia manager Ryne Sandberg said. "It was a carry-over game on the offensive side."
Cincinnati starter Tony Cingrani (2-3) returned from the disabled list and gave up three home runs in six innings.
After the Reds went up 2-0 in the first on Brandon Phillips' RBI groundout and Mesoraco's run-scoring double, Rollins hit his 46th career leadoff homer. Nieves followed with a shot to left that tied the game.
It was the fifth time the Phillies began a game with consecutive homers, the last coming on Sept. 9, 2004, at Atlanta when Rollins and Placido Polanco accomplished the feat.
Chase Utley's fifth-inning groundout scored Lee and put Philadelphia ahead for good. Byrd opened the sixth with a homer to right to push the lead to 4-2.
That was enough offense for Lee, who had received very little support at home this season. He entered 0-3 in four home starts despite a 1.55 ERA.
Lee gave up two runs and nine hits in 6 2/3 innings with three strikeouts and a walk. He was happy to get some runs for a change.
"It's definitely easier to pitch with a lead," Lee said. "It's easier to attack the strike zone, and you can use a little more of the plate and not worry so much if you give up a solo home run or whatever. It's not going to hurt you as bad. If it's a close game or a low-scoring game, every pitch is important. Not that it's not with a lead, but there's definitely more room for error and you can get away with more mistakes."
Without Votto, who remained in Cincinnati for treatment during the series for a distal quadriceps strain in his left knee, and slugger Jay Bruce (knee), who went on the disabled list May 6, the Reds were without two key pieces that might have helped in a comeback bid.
"The elephant in the room is the fact we have Votto and Bruce unavailable," Reds manager Bryan Price said. "Those are pretty essential pieces. That being said, we do have players getting an opportunity to play and do some things. I was very, very optimistic going into today's game that we were going to do some really good things offensively. We got some things started but we weren't able to finish."
Price said there is no timetable for Votto's return and wasn't sure whether the four-time All-Star and 2010 NL MVP would make the trip to Washington, where Cincinnati begins a three-game series Monday night.
The Phillies took two of three from the Reds and improved to 7-0-1 in home series against Cincinnati since 2006.
Byrd upped his team-leading RBI total to 24 with a run-scoring single in the seventh. Asche followed him and broke the game open with a three-run homer to right off Manny Parra that gave Philadelphia an 8-2 lead.
"I think good at-bats are contagious, up and down the lineup. You saw that the last two days," Asche said.
Cingrani allowed four runs and seven hits while striking out seven and walking three. Cingrani, who hadn't pitched since April 30 because of left shoulder tendinitis, had given up three homers all season in 32 1/3 innings before Sunday.
"I felt pretty good today," Cingrani said. "Everything was coming out well. I was spotting it pretty much all day."
The Phillies improved to 11-3 in Saturday and Sunday games. They are 8-19 during the week. ... To make room for Cingrani, the Reds optioned catcher Tucker Barnhart to Triple-A Louisville. ... Reds RHP Mike Leake (2-3, 3.09 ERA) opposes Nationals RHP Stephen Strasburg (3-3, 3.48) on Monday. ... After a day off Monday, the Phillies travel to Miami for three games against the Marlins, with Philadelphia RHP A.J. Burnett (2-3, 3.13) facing Marlins RHP Anthony DeSclafani (1-0, 3.00) on Tuesday. ... Phillies CF Ben Revere was available to pinch-hit. Revere missed games Friday and Saturday due to a stomach virus.