Cozart drove in two runs with a tiebreaking single, and the surging Reds scored five times with two outs in the 11th inning to beat the San Francisco Giants 7-3 on Saturday night.
"I think we're starting to build some momentum, get some confidence," Cozart said. "[Phillips)] had the big home run, birthday boy over here. It was an all-around good game. We played good defense and we had some big hits at the end."
Phillips, celebrating his 33rd birthday, hit a go-ahead homer in the ninth for the Reds before pinch hitter Buster Posey tied it with an RBI double off hard-throwing closer Aroldis Chapman in the bottom half.
"During the game, everybody was like, 'Brandon, do something for your birthday, do something for your birthday,'" Phillips said. "I was like, 'I'm going to try.' I didn't really think I was going to hit a home run off that cat. I don't know what happened. I just tried to make contact. Happy birthday to myself."
Phillips had a couple of bags full of cupcakes and a birthday cake in front of his locker, ready to be devoured.
San Francisco had a great chance to win, with runners at second and third and nobody out. Chapman, however, pitched out of the jam by getting two grounders with the infield in and a strikeout.
The National League West-leading Giants have dropped three in a row and 14 of 18.
After a tight pitching duel between Alfredo Simon and Matt Cain, the Reds trailed 1-0 going into the ninth. Joey Votto drew a leadoff walk from Sergio Romo, and Phillips homered to left field on a 2-2 delivery.
"It started with the walk," Giants manager Bruce Bochy said. "That's the last thing you want to start an inning. He threw some good pitches and then made a mistake to a good hitter."
It was Romo's fifth blown save in 27 opportunities. Opponents had nine hits in their previous 18 at-bats against him entering the game.
"It was too good of a pitch to hit," Romo said. "When you give him a chance like that, he's going to hurt you."
Pitching for the fourth straight day, Chapman couldn't hold the lead. He was handed his second blown save in 18 chances.
Jonathan Broxton (4-0) pitched a scoreless inning for the win.
Simon allowed one run and three hits in seven innings for the Reds. He walked one and struck out four.
Cain gave up six hits in seven innings. He walked one and struck out four.
Cain snagged a hard line drive off the bat of Ryan Ludwick to end the second, stranding a runner in scoring position.
Cain also escaped trouble when he caught Hamilton, who had three hits, in a rundown between second and third to end the first. The right-hander gave up 17 earned runs over his previous 17⅓ innings (8.83 ERA) and has won at home once in his past 12 starts.
Jay Bruce singled twice and had a sacrifice bunt in the 11th for Cincinnati.
Todd Frazier hit into a double play with two on to end the third. He went hitless in six at-bats, snapping a 14-game hitting streak against the Giants.
Thanks to an official scoring change, Phillips had an error taken away from Friday night's game. His errorless streak remains intact at 69 games. Sandoval was credited with an infield hit on the play. ... Chapman had pitched four days in a row once before. ... Giants 1B Brandon Belt (thumb) was transferred to Triple-A Fresno from Class A San Jose two days early on his rehab assignment. ... Giants OF Angel Pagan (back tightness) took a turn on the treadmill and hit off a tee. There's a chance he'll take batting practice Sunday. ... RHP Tim Hudson (7-4, 2.62 ERA) pitches for the Giants in Sunday's series finale. He's lost each of his past two starts. RHP Homer Bailey (7-4, 4.80) goes for the Reds. He's 3-0 in seven games against the Giants. ... Reds RHP Sam LeCure worked a scoreless eighth in his 200th career appearance.