Wilfried Zaha's first-half effort decided a largely uneventful contest, but the Red Devils still managed to finish sixth in the table, and claim a Europa League spot for next term, as seventh-placed West Ham United lost 3-1 at Brighton & Hove Albion.
Captain Harry Maguire was also back in the Man United side following their 4-0 defeat at Brighton & Hove Albion last time out, with Raphael Varane dropping to the bench, but there was no sign of Paul Pogba, who has almost certainly played his final match for the club.
Cavani had the first half-chance in the 16th minute of the contest, but the striker's effort was expertly kept out by returning Palace goalkeeper Vicente Guaita.
The visitors again came close in the 25th minute when Anthony Elanga broke down the right before looking to pick out Bruno Fernandes at the far post, but the Portugal international could not make contact.
David de Gea had to be alert twice in quick succession before the 30-minute mark, meanwhile, first keeping out Zaha's low drive before preventing an effort from Jeffrey Schlupp from flying into the net.
Palace did make the breakthrough in the 37th minute, though, when Zaha worked his way into the penalty box before expertly finding the bottom corner of the net.
Man United had the first attempt of the second period, with Fernandes's long-range free kick kept out by Guaita, as the Red Devils looked to answer back at Selhurst Park.
Conor Gallagher then swept wide of the Man United post just before the hour, with the Eagles still leading 1-0 with 30 minutes to play on the final matchday of the 2021-22 Premier League campaign.
Man United came close to a leveller in the 73rd minute when a cross from Mata was headed back towards his own goal by Joachim Andersen, but the ball dropped just over the crossbar.
The performance was again disappointing, but the result mattered little, as Brighton's victory over West Ham means that Ten Hag will lead the 20-time English champions into the group stage of next season's Europa League, but there is plenty of work for the Dutchman to do ahead of the 2022-23 campaign.
Man United finish the season with a record of 16 wins, 10 draws and 12 defeats from their 38 league games, with a total of 58 points seeing them finish 11 points behind fifth-place Arsenal.