LaRoche lifted a high fly ball into the right-field seats for this 18th homer. Medlen (10-11) was pressed into service after Atlanta exhausted its bullpen.
Washington starter Stephen 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.
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.