Hudson went the distance, tossing his 25th complete game and 13th shutout.
The margin of defeat is the largest for Miami this year. The only game in which the club allowed more runs was on May 24 in a 14-7 loss to the Giants.
"Pitching dictates how the game is going to be," manager Ozzie Guillen said. "Our pitching wasn't there today. We just left a lot of pitches up, and they were ready to swing. Anibal, he's been pitching great for us all year. Today, he was very off. His breaking ball wasn't there. Location, his pitches were high."
Miami had won five of its last six before being shut out for the sixth time this year.
"Game like that, you just have to turn the page as quick as you can and be ready for tomorrow," Guillen said. "I think this ballclub has been playing great. Today was one of those games where you need to shower, go home, and be ready for tomorrow."
Sanchez (3-4) entered with a 2.57 ERA. But it rose to 3.19 after his roughest outing of the season. In his career, he is now 4-11 with a 5.42 ERA against Atlanta.
"The main thing is, I'm healthy," Sanchez said. "That's baseball. The rest of the story, everybody knows. I missed a lot of pitches. I think I was behind for a lot of the counts. But I'm fine."
Hudson, meanwhile, continued to have his way with the Marlins. The veteran is now 12-4 with a 2.76 ERA lifetime against Miami.
The Marlins entered the night tied with Washington for first place in the National League East, but they had a rough opener to their nine-game homestand and fell one game behind as the Nats edged the Mets, 7-6, in 12 innings.
In the third, Jose Reyes extended his hitting streak to 15 games, dropping his third bunt single of the season.
Uggla, who is the Marlins' all-time home run leader with 154, has 200 in his career. The veteran second baseman also has hit home runs in 26 different parks. He also has the distinction of hitting more home runs (78) than any player at Sun Life Stadium, the Marlins' home for 19 seasons.
Uggla put an exclamation point on his big night with a two-run homer in the ninth. The blast off Chad Gaudin was estimated at 432 feet, and it clanked off the 73-foot home run sculpture in left-center.
"It's a cool accomplishment and a cool feat," Uggla said of No. 200. "I hope to have many more."
A borderline pitch that didn't go Sanchez's way ended up being costly for Miami in the fourth. With two outs, Sanchez got two quick strikes on Uggla. The third pitch, a slider, was a called a ball by home-plate umpire Gary Cederstrom. After throwing the pitch, Sanchez took a couple of steps toward the dugout, thinking it was strike three.
Did Sanchez think he struck Uggla out?
"Ask the umpire," he said. "I'm pretty sure, 0-2, it was, 'Strike him out.'"
Uggla worked the count full, and then belted a home run to left.
The Braves weren't through as they strung together three straight singles, by Freddie Freeman, Jason Heyward and Andrelton Simmons, to produce the second run of the inning, giving Atlanta a 3-0 lead.
In the second inning, Uggla hurt the Marlins ahead, setting up their first run. Uggla doubled to left and moved to third on Freeman's groundout to third.
Sanchez executed a good pitch and got Heyward to break his bat. But Heyward rolled a grounder. Although an out was made a first, Uggla scored.
"It just so happens that I've caught a couple on the barrel," Uggla said of his big night.
While Uggla has hit safely in 14 of his 21 games against the Marlins, he considers his former team a tough matchup.
"Even when I was playing behind them I always thought, 'Man, I'm glad I don't have to face these guys on a regular basis,'" Uggla said. "Now I do and it's no fun. You really have to get focused because they have so many weapons they can beat you with."