Kevin Frandsen hit a tying RBI triple off Jonathan Broxton in the eighth, but the NL Central-leading Reds went up to stay on Cozart's shot off Papelbon (3-5).
Cliff Lee took a shutout into the seventh before allowing three runs to remain winless at home this season.
The Reds took a 4-3 lead when Todd Frazier connected off Antonio Bastardo with two outs in the eighth for his 18th homer. The Phillies answered against Broxton (2-1). Domonic Brown hit a two-out single, and Frandsen tripled down the right-field line.
Reds starter Homer Bailey gave up three runs and seven hits in 6 1-3 innings. Aroldis Chapman finished for his 30th save in 34 tries. He struck out Chase Utley swinging at 102 mph fastball with the tying run at third to end it.
The five-time defending NL East champion Phillies had won 12 of 13 against the Reds. The Phillies failed to move within seven games of .500 for the first time since they were 36-43.
A sellout crowd of 45,091 fans each received a Hunter Pence bobblehead. Pence was traded to the San Francisco Giants on July 31, but the Phillies stuck with the promotion because the figurines were already made.
After Lee pitched out of a bases-loaded, no-out jam to preserve a 1-0 lead in the sixth, the Reds rallied in the seventh.
Scott Rolen led off with a double. After Frazier walked, Ryan Hanigan singled up the middle to drive in Rolen. Both runners advanced on an error when first baseman Ryan Howard dropped the ball trying to cut off the throw home.
Lee then struck out Bailey, but Cozart hit a sacrifice fly to put the Reds up 2-1. Drew Stubbs followed with an RBI single, ending Lee's night.
A frustrated Lee jogged off to a standing ovation, but slammed his glove down in the dugout.
Lee allowed two earned runs and nine hits, struck out nine and walked one.
The offense got Lee off the hook in the bottom half.
Frandsen hit a one-out single and Erik Kratz followed with a double to chase Bailey. Sean Marshall entered and walked pinch-hitter Ty Wigginton.
Jimmy Rollins then lined a two-run double to tie it at 3. Juan Pierre got down a bunt, but Wigginton was thrown out at the plate despite crashing into catcher Ryan Hanigan.
Outstanding defense helped Lee in the sixth after three straight singles loaded the bases. Third baseman Frandsen made a diving grab on Ryan Ludwick's grounder in the hole, spun around and threw home from his knee to get a forceout.
The crowd showed their appreciation for Frandsen with a standing ovation.
Hot-hitting Jay Bruce then bounced into a double play, started by second baseman Utley. Rollins made an excellent turn with Ludwick coming in hard. That drew an enthusiastic fist pump from the stoic Utley, who hardly ever shows any emotion on the field.
NOTES: The Phillies are 20-11 against the NL Central, the only division they have a winning record against. ... Lee walked his first batter in four starts in the seventh inning. ... The Reds are 25-11 without 1B Joey Votto. ... Bruce was 0 for 4, snapping his nine-game hitting streak. ... Philadelphia's Vance Worley (6-8, 4.11 ERA) faces Bronson Arroyo (9-7, 3.96 ERA) in a matchup of right-handers Wednesday night.