Washington finished 6 of 13 for 185 yards passing and ran 39 yards for a late touchdown for the Yellow Jackets (8-3, 5-3 ACC).
They rolled up 549 total yards, scored on four straight possessions and held off Duke's late charge to win for just the second time in 1 1/2 months.
Sean Renfree matched a career high with four touchdown passes for the Blue Devils (3-8, 1-6). His 10-yard TD to Cooper Helfet - his second of the day to the tight end - pulled Duke within seven points with 6:37 left.
Duke forced a fumble from Charles Perkins and took over at its 34 with 3:44 left. Renfree's deep pass for Helfet was intercepted by Jamea Thomas with 2:20 left.
Washington and Georgia Tech then ran out the clock to seal their eighth straight win in the series.
Washington showed once again just how capable he is of coming up with big plays both with his feet and his arm. He broke a pair of tackles during his long scoring run, which came 3 minutes after Duke had pulled within 31-24. Four of his completions covered at least 29 yards.
Sims had scoring runs of 4 yards and 1 yard, while Roddy Jones and Embry Peeples had 1-yard touchdown runs for the Yellow Jackets, who entered having lost three of four - including an 11-point loss to Virginia Tech that took them out of the Coastal Division race.
Georgia Tech looked to have taken control of this one by scoring on its final four possessions of the first half. None of those drives took longer than 5 1/2 minutes.
Sims powered in from 4 yards out to tie it at 7. One series later, Washington hit Tyler Melton for 31 yards and raced 21 yards to set up Jones' short scoring run that put the Yellow Jackets ahead to stay.
Washington's 46-yard pass to Embry Peeples then set up Sims' second TD run to make it 21-7 barely 3 minutes into the second quarter. After Duke pulled within a touchdown, Washington lofted a 56-yard pass to Stephen Hill that set up Peeples' TD on the final play of the half to make it 28-14.
Duke wasted no time jumping out to a 7-0 lead, with receiver Donovan Varner taking a sweep 39 yards through the Georgia Tech defense to the 20. Three plays later, Renfree found Vernon over the middle for the receiver's sixth TD this season. His touchdown pass to Varner on fourth-and-3 pulled the Blue Devils to 21-14 with 5:24 left before halftime.
Will Snyderwine kicked a 35-yard field goal 5 1/2 minutes into the third quarter to make it 28-17. Justin Moore countered with a 25-yard field goal with just under 2 minutes left in the third to put the Yellow Jackets back up by 14.