Smith completed 31 of 46 passes for 296 yards and a score in the 23-17 win on Monday at Heinz Field in Pittsburgh.
Washington held Pittsburgh to just 21 rushing yards in the victory. Washington linebacker Jon Bostic intercepted Steelers quarterback Ben Roethlisberger with two minutes remaining to ice the victory.
"We knew this was going to be kind of be this brand of football coming up here," Smith told reporters. "Playing Pittsburgh in December, we knew it was going to be a little bit ugly.
"But proud of our team I think to find a way to win this game like that, especially as young as we are, I think it showed a lot of maturity."
Neither team scored in the first quarter. Roethlisberger then connected with wide receiver Diontae Johnson on a three-yard touchdown toss to start the second quarter.
Both offenses continued to struggle until Roethlisberger threw a 50-yard touchdown pass to James Washington to give the Steelers a 14-0 lead.
Washington then started its rally.
Dustin Hopkins made a 49-yard field goal at the end of the first half to make the score 14-3 at the break. Washington then began the second half with a 14-play, 95-yard touchdown drive. Running back Peyton Barber ended the drive with a one-yard touchdown run.
The Steelers pushed their lead back to seven points when kicker Matthew Wright made a 37-yard field goal at the start of the fourth quarter.
Washington later tied the score when Smith threw a 15-yard touchdown pass to tight end Logan Thomas. Hopkins added two field goals down the stretch to secure the victory.
Roethlisberger completed 33 of 53 passess for 305 yards, two scores and an interception in the loss. T.J. Watt had five tackles, two tackles for a loss, a sack and a forced fumble for the Steelers. Thomas had 98 yards and a score on nine catches for Washington.
"These guys deserve it," Washington coach Ron Rivera said. "We have been down for such a long time and we're trying to rebuild ourselves and build up.
"This is something that we can build off of and just really proud of the way they played."
The Steelers (11-1) face the Buffalo Bills at 8:20 p.m. EST on Sunday in Orchard Park, N.Y. Washington (5-7) faces the San Francisco 49ers at 4:25 p.m. EST on Sunday at State Farm Stadium in Glendale, Ariz.
"We step into a stadium each week with intentions towards winning," Steelers coach Mike Tomlin said. "Our intentions will be the same next week. We just live one week at a time. It's not a collective for us."