He started Randy Foye in place of Hayward and Derrick Favors for Marvin Williams.
The move gave Utah a starting front court that included 6-foot-10 Al Jefferson at center, the 6-10 Favors at power forward and 6-foot-8 Paul Millsap at small forward.
The Jazz didn't play well early against the winless Wizards, falling behind by 13 points in the opening minutes. But Hayward came off the bench to fuel a comeback and the Jazz finished with a 2-2 split on their four-game road trip.
"I thought it was average," Hayward said. "I thought we definitely could have played better. ... But it wasn't a disaster, going 2-2. I think you've got to be happy with that."
The Jazz have already played an NBA-high eight road games, but three of their next four are at home, where Utah is 3-0.
"It feels like we've been gone forever, so it's nice to be back for the holidays," Hayward said. "I know a lot of people have family coming in. ...
"We need to get back to Utah Jazz-style of basketball in front of our fans. Our crowd is going to be good for us."
Hayward finished with 15 points and five assists in 25 minutes in his first non-start of the season at Washington.
Asked if he was O.K. with the new role, Hayward nodded and said, "Whatever coach wants us to do, I'm going to do. As long as we get wins, that's what it's all about."