Strasburg was ejected in the second inning. He hit Justin Upton with a pitch in the first and was tossed after throwing two straight pitches behind Andrelton Simmons' back.
LaRoche lifted a high fly ball into the right-field seats for this 18th homer off Kris Medlen (10-11), who was pressed into service after Atlanta exhausted its bullpen.
The game lasted 5 hours, 29 minutes.
Jason Heyward hit two homers, including a two-run shot in the ninth as Nationals closer Rafael Soriano blew a 7-5 lead.
Craig Stammen (6-5) allowed no hits in three scoreless innings, striking out five with one walk. Dan Haren pitched the 15th for his first career save.
Atlanta's Brian McCann had three hits, including a two-run homer. Freddie Freeman also homered for the Braves. Ryan Zimmerman hit a homer for Washington.
Each team used all its relievers.
Medlen, scheduled to start Tuesday at the Mets, made his first relief appearance of the season as Atlanta's ninth pitcher of the night.
Soriano suffered his sixth blown save. Heyward's second homer of the game, a two-run shot to right-center with two outs in the ninth, sent the game to extra innings.
Braves pitchers hit Bryce Harper with two pitches in Atlanta's win on Friday night, leaving the outfielder bruised and unable to start on Saturday night.
That set the stage for Strasburg.
Strasburg hit Upton in the first inning one pitch after Heyward's leadoff home run. Strasburg was tossed by plate umpire Marvin Hudson after throwing two straight pitches behind Simmons' back in the second.
Hudson also tossed Nationals manager Davey Johnson when he came out to argue.
The drama left Braves fans howling, but no players emerged from either dugout. There were no additional incidents in the game.
Entering the game, Strasburg had thrown three wild pitches in 146 1-3 innings. He doubled that total by unleashing three consecutive wild pitches, including the two behind Simmons' back.
There were tensions between the teams that carried over from the Braves' three-game sweep at Washington on Aug. 5-7. Both benches emptied when Harper was hit by a Julio Teheran pitch in the second game of the series.
Tanner Roark dominated the Braves after taking over for Strasburg. Roark allowed only one hit in four scoreless innings. He had a run-scoring double in the fifth for his first major league hit and RBI.
Mike Minor allowed four runs in only 1 2-3 innings, the shortest of his 78 career starts.
Minor gave up bases-loaded, two-run singles to Wilson Ramos in the first and Jayson Werth in the second.
Harper was in Johnson's original lineup, but was held out with bruised triceps until entering the game as a defensive replacement in the ninth.
Strasburg gave up two runs and two hits, two walks and one hit batter in one-plus inning. Like Minor, he also had the shortest start of his career.
Werth doubled off Luis Ayala in the sixth and scored on a single by Wilson Ramos.