Ryan Howard's three-run homer off Hudson during a five-run fifth tied it at 6, but the wild card-leading Braves scored four runs after Antonio Bastardo (2-4) got the first two outs in the seventh.
Michael Bourn had three hits, including a two-run homer, and drove in three, and Uggla also had three hits.
Chad Durbin (4-1) tossed a perfect sixth to earn the win.
Hudson allowed six runs and six hits in 4 1-3 innings. It was his second-shortest outing of the season.
The Braves chased Phillies starter Kyle Kendrick in the fourth, taking a 6-1 lead that seemed insurmountable with Hudson on the mound. The three-time All-Star righty is 132-3 when given a lead of three or more.
But Jimmy Rollins got a rally started in the fifth with an RBI double. Domonic Brown followed with an RBI double to cut it to 6-3. After Chase Utley was hit by a pitch, Howard took Hudson deep to left-center to tie it at 6.
The comeback took Kendrick off the hook. He gave up six runs and seven hits in 3 1-3 innings.
Jason Heyward's two-out double in the seventh started Atlanta's rally. Chipper Jones was intentionally walked and Freddie Freeman worked a walk. Uggla then ripped one down the third-base line to clear the bases and knock out Bastardo.
Uggla entered the game hitting just .136 (23 for 169) since June 6. However, the All-Star second baseman has 11 hits and 11 RBIs in the last 10 games.
The Braves added another run in the inning off Josh Lindblom and two more in the eighth.
Rollins hit his 40th career leadoff homer, but the Braves answered with four runs in the second.
Freeman walked and Uggla singled with one out. Both advanced on a wild pitch that proved costly because David Ross followed with a grounder to shortstop that could've been an inning-ending double play. Instead, Freeman scored on the out.
After Paul Janish walked, Hudson hit a drive over center fielder John Mayberry Jr.'s head to knock in a pair. Kendrick then walked Bourn and Martin Prado singled to right. Hudson was safe at the plate when catcher Erik Kratz dropped the throw by Nate Schierholtz.
Bourn connected in the fourth to extend Atlanta's lead to 6-1.
The last-place Phillies lost their first series since trading Shane Victorino and Hunter Pence on July 31. The five-time defending NL East champions fell to 50-61. They lost just 60 games last year.
Notes: Rollins is fifth on the all-time list for leadoff homers, four behind Brady Anderson for fourth place. ... Howard is 20 for 61 with seven homers and 17 RBIs. ... The Phillies had a 1.45 bullpen ERA since July 20 before allowing five earned runs to the Braves. ... The Phillies are 3-9 vs. the Braves this season. ... Atlanta's bullpen has a major league-best 1.59 ERA since July 3. ... The Braves are 41-13 in games in which Bourn has a hit and scores. ... Both teams are off Thursday. Paul Maholm (9-7, 3.75) takes the mound when the Braves visit the New York Mets on Friday night. ... Roy Halladay (5-6, 4.02) pitches against St. Louis in a rematch of Game 5 of the 2011 NLDS that he lost 1-0 to Chris Carpenter.