The Blues led through Nicolas Jackson at Stamford Bridge, but Eddie Howe's side levelled before half time through frontman Alexander Isak, who converted Lewis Hall's cross from close range.
But the influential Palmer's fizzing 47th-minute drive won what was a free-flowing contest for the hosts, who move five points clear of the Magpies in the Premier League table - with the two sides now set to meet again on Tyneside on Wednesday in the Carabao Cup fourth round tie at St. James' Park.
In an open, fast-paced start, Noni Madueke tested Nick Pope after a Jackson break, and just four minutes in another Jackson run cut through the United backline. The Senegalese forward turned Dan Burn and played in Palmer, whose low shot went in off the upright, but the goal was ruled out for offside following a VAR check.
Just a couple of minutes after Levi Colwill was adjudged not to have tripped Bruno Guimarães just outside the box as he looked to latch on to Isak's through ball, the hosts took the lead - and this time it stood. Palmer helped create it with a magnificent angled pass in behind Tino Livramento to free Pedro Neto. The winger then skipped past Fabian Schär and sent over a low cross that Jackson converted with ease, his firm finish beating Pope.
Miguel Almirón - who was brought into Newcastle's starting 11 along with Harvey Barnes at the expense of Jacob Murphy and Anthony Gordon, who missed out with a groin issue - scuffed a half-chance wide soon after, while at the other end Neto forced another stop from Magpies custodian Pope.
But United enjoyed a more promising spell in the latter stages of the half and it led to a leveller. Barnes and Sandro Tonali saw efforts blocked before Almirón's was held by Robert Sánchez, and just after the half-hour mark they carved out an equaliser. Livramento's industry in the middle helped create it, with Barnes feeding the overlapping Hall on the left, and the former Chelsea youngster's cross was turned home with typical coolness by Isak.
It was Isak's second goal of the season and was a reasonable reward for Howe's side's improvement as the half wore on, though Pope did have to spring to his right to keep out a deflected Neto drive before the break. They couldn't maintain that momentum in the second half's early exchanges, though, and they fell behind again just over a minute after the restart. Pope may feel he could have done better with Palmer's powerful strike from outside the area, but it fizzed beyond him at the near post to make it 2-1.
Moments later it could easily have been three, with Neto climbing above Livramento to meet Madueke's deep cross and seeing his header cannon back off the post.
Howe responded by making three changes with just under 25 minutes remaining, introducing Sean Longstaff, Joe Willock and Jacob Murphy in place of Barnes, Tonali and Almirón just after Fabian Schär whipped a free kick just over the bar. The changes buoyed the visitors, as Willock's cross was met by Isak, who was denied a second equaliser of the afternoon by Reece James, who blocked the attempt almost on his own goalline.
Isak then had another chance with quarter of an hour remaining, as he rounded goalkeeper Sánchez after shrugging off Wesley Fofana. The backtracking Colwill blocked his route to goal but after trying to sidestep the England international, Moises Caicedo was able to retreat and get a challenge in before Isak could shoot or square to the unmarked Joelinton, snuffing out the opportunity.
In the end, Chelsea went close to a third despite some pressure from Newcastle, with substitute Mykhailo Mudryk firing straight at Pope inside the final ten minutes. In the final minute, referee Simon Hooper awarded a penalty after Christopher Nkunku was fleetingly pulled back by Burn but overturned his own decision following a VAR review, and United slipped to defeat.
