Turner's scoring play came with 2:46 remaining and boosted the Falcons to their first 5-0 start in franchise history.
Matt Ryan amassed 345 aerial yards with two touchdowns and an interception, completing 34-of-52 passes. His favorite target was Tony Gonzalez, who made 13 grabs for 123 yards and a touchdown.
Julio Jones chipped in with 10 catches for 94 yards and a score.
Washington quarterback Robert Griffin III was limited to 10-of-15 through the air for 91 yards before leaving the game after Falcons linebacker Sean Weatherspoon delivered a blow to his head.
Kirk Cousins came in and tossed for 111 yards with a TD but was intercepted twice for the Redskins (2-3), who dropped their eighth straight home game.