With Tottenham held by Watford earlier in the day and Manchester United facing runaway leaders Liverpool on Sunday, Chelsea would have cemented their grip on fourth place with a win on Tyneside.
But Chelsea were unable to muster the cutting edge required to break down stubborn Newcastle and fell to Hayden's header with just 20 seconds left in stoppage-time.
Chelsea's first defeat in four league matches leaves them five points clear of fifth-placed United.
After four league games without a victory, this was a much-needed success for Newcastle, but a wasted opportunity for Chelsea to open up a bigger gap in the race to qualify for the Champions League.
Newcastle are reportedly in talks over a loan deal for Inter Milan winger Valentino Lazaro and their two current wide men, their places in danger, combined to fashion the hosts' first chance.
Miguel Almiron opened up Chelsea down the right flank and Allan Saint-Maximin met the cross with a header that looped over.
Newcastle's lengthy injury list gained another casualty inside 12 minutes when Jetro Willems was stretchered off after taking a knock in a challenge on Callum Hudson-Odoi.
Chelsea were dominating possession but Newcastle striker Joelinton, his confidence lifted by an FA Cup goal against Rochdale in midweek, went close to his first league goal since August when his powerful header cannoned back off the bar.
Reece James signed a new contract with Chelsea this week as a reward for his impressive breakthrough campaign and the young right-back gave a glimpse of his potential when his pin-point pass found N'Golo Kante for a shot that Martin Dubravka saved with his leg at the near post.
Chelsea remained vulnerable at the back and Federico Fernandez headed wastefully over from a Jonjo Shelvey free-kick.
Willian squandered a good opportunity from a Chelsea counter early in the second half when the Brazilian dragged wide from the edge of the area.
Chances were at a premium and Cesar Azpilicueta's long-range effort lacked the power to beat Dubravka before Shelvey finally had Newcastle's first shot on target in the 59th minute.
Chelsea's frustration mounted when James limped off injured and moments later Tammy Abraham, teed up by Azpilicueta's header, saw his effort turned away by Dubravka's brave save.
Emerson fired just wide with an angled drive and in the fourth minute of stoppage-time Chelsea paid for their profligacy.
Saint-Maximin had space to whip in a teasing cross and Hayden got behind Antonio Rudiger for a close-range header that keeper Kepa Arrizabalaga could only push into the net.