PHILADELPHIA -- Bryce Harper should make more predictions.
The Phillies are 9-1 since Harper said the team needs to "win nine of 10" following a loss at Atlanta on Aug. 22.
"I heard someone say Bryce is a prophet," Phillies manager Joe Girardi said. "I'm going to go to him and see what I should do next."
Philadelphia improved to three games over .500.
"Being able to go out and do what we've done, what they've done because I feel like I haven't done anything, we've played to the best of our ability and we still can get better," said Harper, who has been slumping.
The defending World Series champion Nationals have lost six straight and nine of 10 to fall to 12-23.
"It was one of those losses where you scratch you're head and just kind of forget about it and come back tomorrow," Nationals manager Dave Martinez said. "We had our chance there. When you battle in a game that close every opportunity mattered, so you've got to drive in that run."
Didi Gregorius sacrificed J.T. Realmuto to third to start the bottom of the 10th inning against Sean Doolittle (0-2). After Jean Segura was intentionally walked, he took second base on defensive indifference. Pinch-hitter Phil Gosselin was then intentionally walked to load the bases.
Bohm hit a shallow fly to center field but Realmuto slid home well ahead of Victor Robles' throw. Bohm also had three singles, all to the opposite field.
"He's a hitter," Girardi said of the rookie third baseman. "I think his approach is great. He likes to hit the ball gap to gap."
It was Philadelphia's first four-game sweep against the Nationals since May 15-17, 2009, at Washington. The Phillies hadn't swept a four-game series against the Nationals franchise in Philadelphia since August 9-12, 1991, when they swept the Montreal Expos at Veterans Stadium.
Nationals starter Anibal Sanchez allowed four runs -- one earned -- and eight hits in 3 1/3 innings.
Taylor connected off David Phelps in the seventh, giving the Nationals a 5-4 lead on his fourth homer and second in the series.
The Phillies took a 2-0 lead in the first when shifted third baseman Brock Holt couldn't handle Gregorius' sharp one-hopper for an error, allowing two runners to score.
Turner tied it at 2 in the third when he circled the bases after his drive hit off the right-center field wall and bounced back toward left-center. He scored easily on a headfirst slide with no play at the plate.
"We did a lot of things right," Turner said. "So it's what you or whoever wants to focus on. Do you want to be positive or do you want to feel bad for yourself? So I think we did a lot of good things and if we play like we did tonight I think we should win more often."
Rhys Hoskins launched his seventh homer in the bottom half and Gregorius added an RBI double to give Philadelphia a 4-2 lead.
Turner's RBI groundout and Juan Soto's RBI single tied it at 4 in the fifth.
DICK ALLEN DAY
The Phillies retired Dick Allen's No. 15 in a ceremony before the game. The 78-year-old Allen was a seven-time All-Star, 1964 National League Rookie of the year and 1972 American League MVP. He batted .292 with 351 homers, 1,119 RBI and .912 OPS in 15 seasons, including nine with the Phillies over two stints.
Nationals: RHP Austin Voth (0-4, 7.99 ERA) starts the opener and RHP Will Crowe (0-1, 4.91) pitches the second game of a doubleheader at Atlanta on Friday.
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