The Reds won for only the second time in 12 games.
Strasburg gave up a solo homer in the first inning to Joey Votto and walked Brayan Pena on his 16th pitch to open the second. He shrugged after the pitch, a trainer went to the mound, and Strasburg headed for the clubhouse after a brief conversation.
Taylor Jordan (0-1) took over and gave up three runs in the sixth. Brandon Phillips — limited to pinch hitting because of a sore toe — tied it with a sacrifice fly. Billy Hamilton singled for a 3-2 lead, stole second, and scored on Cozart's third single of the game.
Anthony DeSclafani (3-4) went six innings for his first win since April 20.