Cliff Lee (4-4) struck out three in 6 2/3 innings for his first home victory of the season.
Cincinnati's Devin Mesoraco had a career-high four hits, including two doubles and an RBI. The Reds again played without the injured Joey Votto, who remained in Cincinnati during the three-game series with a strained quad.
Philadelphia's offense came to life 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 12 runs in Saturday's victory, which snapped streaks of four straight losses and 23 consecutive scoreless innings.
After the Reds went up 2-0 in the top of the first Sunday on Brandon Phillips' RBI groundout and Mesoraco's run-scoring double, Rollins hit his 46th career leadoff homer. Nieves followed with a solo shot to left to tie the game.
It was just the fifth time in club history the Phillies began the game with consecutive homers, the last coming 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 solo homer to right to push the lead to 4-2.
That was enough offense for Lee, who had gotten very little support at home this season. He entered 0-3 in four home starts despite posting a 1.55 ERA. Lee gave up two runs on nine hits with three strikeouts and a walk.
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 deep right off Manny Parra that gave Philadelphia an 8-2 lead.
Reds starter Tony Cingrani (2-3) returned from the disabled list and allowed four runs on seven hits in six innings while striking out seven and walking three. Cingrani, who hadn't pitched since April 30 because of left shoulder tendinitis, had given up a total of three homers all season in 32 1/3 innings before the three Sunday.