Besides Belt, there were plenty of heroes for San Francisco, starting with veteran pitcher Tim Hudson, who threw 7 1/3 innings, allowing just one run and seven hits while striking out eight and walking none.
But Hudson's performance was almost overshadowed by that of the Nats' starter, Jordan Zimmerman, who threw a three-hitter over 8 2/3 innings, including a stretch when he retired 20 consecutive batters from the third through the ninth.
As MLB.com noted, "The Giants were so overmatched that they hit only one ball out of the infield during that stretch. They also advanced one runner as far as second base through the first eight innings."
But then another of the Giants' post-season heroes, rookie Joe Panik, coaxed a walk from Zimmerman, who then was replaced by reliever Drew Storen. He gave up a single to Buster Posey, the catcher's third hit of the game, and a double to Pablo Sandoval, which scored Panik and tied the score 1-1.
In extra innings, neither team could muster any offense for what turned out to be the equivalent of a second complete game, culminating with Belt's blast leading off the 18th.
Notable for the Giants was pitcher Yusmiero Petit, who threw six shutout innings, allowing one hit and three walks while striking out seven.
Hard-throwing rookie relief pitcher Hunter Strickland closed out the bottom of the 18th, the last of 17 pitchers (eight for the Giants, nine for the Nats) to appear in the game.
Update Friday 4:09 p.m. From AP:
Never much of a postseason performer, Jake Peavy sure outpitched Stephen Strasburg in the former No. 1 pick's playoff debut.
Guess that October aura the San Francisco Giants proudly own has rubbed off on Peavy.
The intense right-hander took a no-hitter into the fifth inning, San Francisco's bullpen managed to hold onto a lead, and the wild-card Giants won their ninth consecutive postseason game by beating Strasburg's Washington Nationals 3-2 on Friday in an NL Division Series opener.
Peavy won the 2007 Cy Young Award but was 0-3 with a 9.27 ERA in five previous starts beyond the regular season. This time, the 33-year-old right-hander threw 5 2-3 scoreless innings, allowing only two hits.
Buster Posey, Joe Panik and Brandon Belt drove in San Francisco's runs.
Game 2 is Saturday, with Washington's Jordan Zimmermann - who threw a no-hitter in the regular-season finale - facing Tim Hudson.
Peavy was lifted after his third walk, then screamed and cursed as he stomped toward the dugout with two runners aboard in the sixth. Reliever Javier Lopez walked his only hitter, loading the bases.
Giants manager Bruce Bochy turned to Hunter Strickland, a rookie with all of seven innings on his major league resume. Calm as a 10-year veteran, Strickland took care of Ian Desmond - 8 for 12 with a grand slam and 17 RBIs with the bases full this season - on four pitches: 99 mph ball, 98 mph swing-and-miss, 99 mph called strike, 100 mph swing-and-miss.
But in the seventh, Strickland allowed Bryce Harper's upper-deck homer to right on a 97 mph fastball and, one out later, Asdrubal Cabrera hit pretty much the same pitch over the wall in right, too, making it 3-2.
Reliever Jeremy Affeldt got the last out in the seventh. The Nationals put runners on first and second with one out in the eighth against Sergio Romo, but he struck out Desmond and got Harper on a grounder.
Santiago Casilla pitched a perfect ninth for the save.