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Justin Turner had a two-run single and the Mets scored five runs in the first inning with two outs. Andy Pettitte looked flat after waiting out a rainstorm that set back the start of the season's second Subway Series by 53 minutes.
Davis followed Turner's hit with a fly to right that a leaping Nick Swisher got his glove on. But when he hit the wall, the ball was nudged off the top of the fence and over for a 5-0 lead.
Still, the homer-happy Yankees rallied with three late drives to set up a chance to see Francisco.
Alex Rodriguez hit his 641st homer and Andruw Jones also connected for the Yankees against Jonathon Niese. Robinson Cano added a two-run shot off Miguel Batista in the eighth to cut the lead to 6-4 after David Wright had an RBI single in the seventh.
Francisco made headlines in New York this week by saying "I can't wait to face those chickens" and expressing his desire to strike out the side against them.
He explained Friday afternoon what he meant by his strange comment, saying he thinks the Yankees often complain about umpires' calls.
In a heavy drizzle, Francisco made his first appearance against the Yankees for the Mets and - you got it - played chicken with a powerful lineup. Andres Torres made a stumbling catch in deep center field for the first out. The right-hander walked pinch-hitter Raul Ibanez before Derek Jeter singled. Francisco then struck out Curtis Granderson and got Mark Teixeira to pop out for his 18th save in 21 opportunities.
The Yankees lost their third in a row following a 10-game winning streak that began against the Mets. They had not lost three in a row since May 19-21.
The afternoon rain didn't deter a crowd of 40,191 from filling Citi Field. It did help alleviate the sizzling heat of the last few days that even inspired a Mets ticket offer. The Mets sold a limited number of choice seats for $94, the price of the temperature at the start of Wednesday night's game against Baltimore.
The Mets totaled seven runs when they were swept in three games in the Bronx in early June. In the finale at Yankee Stadium, Niese (5-3) carried a 3-0 lead into the seventh but he was betrayed by his defense and gave up two unearned runs.
This time, he gave up the two homers and eight hits in 6 1-3 innings to help the Mets win their seventh straight interleague game.
Pettitte (3-3) gave up five runs for the second time this season and for the first time in an inning since Sept. 4, 2001, against Toronto.
Torres walked in the first and Ronny Cedeno, activated from the disabled list before the game, singled. Wright hit into a fielder's choice and Lucas Duda flied out. But the Mets lead the majors in two-out runs (155), and they soon cashed in.
Turner hit a two-run single to center. Davis then hit his eighth homer of the year.
Batting just .162 entering the game on June 10, Davis didn't even start with Pettitte on the mound. But the left-handed hitter had a pinch-hit double that tied the score in the ninth that day and has played well enough recently to earn a place in manager Terry Collins' lineup Friday against the crafty lefty.
As has often been the case this season, the Yankees had trouble with runners on base. Leading the majors with 105 homers coming in but hitting only .218 with runners in scoring position, the Yankees put two on in the first and second and had a runner on in the third, fourth and seventh without scoring.
Rodriguez homered with one out in the sixth, his drive landing in the container that holds the big apple in straightaway center. Jones homered to left in the seventh. Cano hit a two-run shot in the eighth.
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