Morgan Rogers and Palmer scored either side of half-time to secure three important points for Chelsea, who responded positively after defeats against Tottenham Hotspur and Liverpool.
Rogers opened the scoring shortly after the interval, producing an excellent individual finish after Romeo Lavia recovered possession in midfield. Palmer subsequently converted a penalty to reach 50 Premier League goals for Chelsea in just his 105th appearance.
Only Jimmy Floyd Hasselbaink and Diego Costa have reached that milestone in fewer appearances for the club.
Palmer was particularly impressed with Rogers' contribution, having watched his team-mate carry the ball almost the length of the Manchester United half before lifting his finish over goalkeeper Senne Lammens.
“That's what he does, you know,” Palmer said.
“He gets the ball between the lines and when he drives he makes things happen. He went all the way himself and it was a great finish.”
There was also considerable encouragement for Estevao, who was rewarded with his first Premier League start of the campaign after struggling to secure regular opportunities under Xabi Alonso.
The Brazilian produced an energetic display on the wing and twice found the net, although both efforts were ruled out for offside. The second decision was particularly marginal, with VAR determining that he had strayed beyond the defensive line by the narrowest of margins.
Palmer believes the teenager demonstrated his quality and deserved the opportunity after remaining patient during a difficult start to the season.
“He needed to be patient for his moment and he got his moment today,” Palmer said. “You could see in moments he was very good.
“He was unlucky with the offside; I think it was literally a centimetre. But we know the talent he's got and we all love him. So hopefully he can continue like that.”
Estevao's performance was particularly significant given the questions surrounding his limited involvement in recent weeks. He had been left on the bench during Chelsea's Carabao Cup elimination against Liverpool, prompting discussion about his position in Alonso's plans.
The victory also provided a timely response to a difficult week, with Chelsea having lost against Tottenham before their midweek defeat on penalties at Anfield.
Palmer acknowledged the importance of returning to winning ways and explained how the team improved after the interval.
“It was very important to get back to winning ways, obviously after the last defeat, so it's good.
“I just think we were more intense. The chances they had were from us losing the ball, so we tried to play a bit smarter, a bit sharper and not losing the ball as easily, and I think it helped.”
Chelsea moved back into the Premier League's top four, at least temporarily, with the result representing their third victory in four league matches since the previous international break.
For Estevao, however, the personal significance could prove just as important. After waiting for his opportunity, he produced enough against United to strengthen his case for further involvement.