The NL Central leaders completed their first three-game sweep in New York since 2001 and matched their longest winning streak of the season. The Reds also finished 6-2 this year in visits to Citi Field and Yankee Stadium.
Cueto (8-3) struck out a season-high eight in seven innings. He also doubled for the first extra-base hit of his career.
The only run against Cueto came when he issued a bases-loaded walk to pitcher Chris Young - at 6-foot-10, the Mets starter has a strike zone as large as anyone in the majors.
Sean Marshall got four outs for his ninth save in 10 chances.
Phillips' single capped a three-run rally in the fifth, helped by right fielder Lucas Duda's wild throw. There was plenty of wildlife on the field, too - a squirrel scampered into the Reds' dugout in the ninth, and several pigeons spent part of the game wandering around the infield dirt.
It was 3-1 in the sixth when Duda led off with a single and Ike Davis followed with a hard grounder up the middle that Phillips backhanded. While in full stride, the Gold Glove second baseman tossed the ball with his bare hand between his legs to shortstop Zack Cozart, who made the DP relay. Phillips broke into a big smile after his latest highlight-reel play.
Young (1-1) gave up two earned runs in seven innings. His four-game winning streak, which began in 2010 and was prolonged by a series of injuries, came to an end.
The Mets lost their sixth straight game on Father's Day. Davis, whose dad, Ron, was a longtime pitcher in the majors, scored New York's only run when Young walked with two outs in second.
Cueto had been 0-3 lifetime against the Mets and had consistently been hit hard by them. He allowed six hits and walked one.
Cueto had gone 216 at-bats at the plate with only singles until he crept up in the batter's box and led off with an opposite-field drive that bounced over the right-field wall in the third. He was left stranded and then in the bottom of the inning, he threw one pitch and needed a break.
Reds manager Dusty Baker and a trainer went to Cueto and eventually a cup was brought to give the right-hander a drink.
Cozart doubled with two outs in the fifth and scored the tying run on a single by Wilson Valdez. Joey Votto was intentionally walked and Phillips hit an RBI single, with another run scoring when Duda's throw to third bounced into the stands for an error.
NOTES: A day after the Mets put LF Jason Bay on the 7-day disabled list for concussions, manager Terry Collins said it was too early to predict his future. "I'm really worried about the severity of this injury," Collins said. "There's always the possibility that he doesn't bounce back." ... Two-time Super Bowl MVP Eli Manning threw out the first ball as part of a promotion to celebrate Father's Day. "I'd like to get up on the mound, toe the rubber, whip it in there," said the Giants QB and former high school 2B. Instead, clutching 15-month-old daughter, Ava, in his left arm, he lobbed a toss to Mets star David Wright from the grass. Manning said he took a few warmups in the bullpen during his first trip to Citi Field. Asked whether he preferred the Mets or Yankees, he smartly answered: "I'm a New York fan." ... Mat Latos (5-2, 4.64 ERA) starts for the Reds on Monday night at Cleveland vs. Derek Lowe. Baker and Lowe had a verbal dustup a few days ago. ... Knuckleballer R.A. Dickey (10-1, 2.20) starts for the Mets on Monday at home against Baltimore's Jake Arrieta.
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