However, there was a manic finish with Sadio Mane and Palace substitute Max Mayer both scoring in stoppage time after Milner had been sent off for a second bookable offence on Wilfried Zaha.
Ole Gunnar Solskjaer racked up a sixth successive league win at the start of his Manchester United reign thanks to Paul Pogba's penalty and Marcus Rashford's strike securing a 2-1 win over Brighton.
On a day of commemorations to mark 25 years since the death of Matt Busby, Solskjaer surpassed the start of the managerial great who achieved five wins in a row when he set out at United in 1945.
"Maybe in 10, 15, 20 years, I'll look back on things like this," he said. "I don't care whatsoever about records. But of course to be part of a team that's winning so much, that feels great."
United remain sixth, just as it was when Solskjaer was hired to replace Jose Mourinho a month ago, but now Champions League qualification is back on the cards. The race for fourth is looking wide open, with fourth-placed Chelsea only three points ahead of Arsenal and United.
Maurizio Sarri had little to celebrate and chose to eviscerate his Chelsea squad following a 2-0 defeat at Arsenal.
"It appears this group of players is extremely difficult to motivate," the Italian coach said after seeing Chelsea manage just one shot on target after Alexandre Lacazette and Laurent Koscielny scored in the first half for Arsenal.
All Whites striker Chris Wood was involved in the pivotal incident of Burnley's 0-0 draw at Watford.
The striker looked to have tapped in a winner from close range in stoppage time but his effort was ruled out for offside, which infuriated his manager Sean Dyche.
"I didn't think it was [offside] at the time, and looking back, I don't think it was now and I'd be surprised if the majority didn't think it was onside," said Dyche. "Unfortunately a big decision has gone against us.
"VAR is a definite for me. We have not always had the right decisions go our way, and we are 65 Premier League games without a penalty, which is an anomaly."
Of the offside call, Watford manager Javi Gracia said: "I didn't see but I always support and respect the referee's decisions."
The point means 16th-placed Burnley have gone four league games unbeaten and move three points clear of the relegation zone.
In other results, Wolverhampton edged a pulsating encounter with Leicester 4-3 when Diogo Jota completed a hat-trick in the third minute of stoppage time.
Newcastle climbed out of the relegation zone and sent Cardiff into the bottom three with a 3-0 victory over the Welsh club. Southampton's revival under Ralph Hasenhuettl continued with a 2-1 victory over Everton taking the south coast club three points clear of danger. In a mid-table meeting, Bournemouth beat West Ham 2-0.
- AP
