The hard-fought home victory took Slavisa Jokanovic's side into second and two points above third-placed Cardiff City, who visit Hull City on Saturday, in the race for the Premier League.
Fulham's Serbia striker Mitrovic, on loan from Newcastle United, struck in the 75th minute with a downward header after Brazilian Piazon had levelled for the hosts before halftime
"Today it was complicated for us and I am happy to take the pressure off our back. It's part of the competition," said Jokanovic. "Aleksandar's contribution is crucial. He's scored many winning goals and today it was an important goal for us."
The home side fell behind just before the half-hour when Sunderland's Joel Asoro celebrated his 19th birthday with a superb goal after turning smartly and firing home a low shot.
The strike stunned the expectant Fulham fans but they were soon cheering when Brazilian Piazon, on loan from neighbours Chelsea, equalised from a Ryan Fredericks cross before Mitrovic rose high above the Sunderland defence to grab the winner.
Fulham, who extended their unbeaten run to 23 games, have 88 points with one game left. Cardiff are on 86 but with two games to play. The top two go up automatically with the next four entering the playoffs for a third promotion place.
Wolverhampton Wanderers have already been promoted as champions, while Friday's defeat confirmed that Chris Coleman's Sunderland will finish bottom of the second-tier after being relegated as the basement side in the top flight last season.
(Writing by Ken Ferris; Editing by Pritha Sarkar)
