Kieran Trippier and substitute Harvey Barnes – making his first appearance since September – hit the crucial goals after the Hatters had taken control early in the second half with a twice-taken Carlton Morris penalty and an Elijah Adebayo effort.
The score had been 2-2 at the break, with Luton twice cancelling out goals from Newcastle's Sean Longstaff thanks to replies from Gabriel Osho and Ross Barkley.
The point was enough to lift Luton above Everton and Nottingham Forest and into 16th place.
They had begun the game in the relegation zone after Everton briefly moved out of the bottom three by snatching a last-gasp 2-2 draw against Tottenham earlier in the day.
Jarrad Branthwaite grabbed the Toffees' stoppage-time equaliser at Goodison Park after Richarlison had returned to haunt his old club with a first-half double for Spurs.
The in-form Brazilian took his tally to nine goals in his last eight league appearances as he netted after just four minutes and again shortly before the interval.
Yet Everton replied to the opener as Jack Harrison was credited with helping a Dominic Calvert-Lewin header in and Branthwaite had the final say with a back-post finish four minutes into added time.
New signing David Fofana scored twice as fellow strugglers Burnley also rallied to claim a 2-2 draw against Fulham.
Joao Palhinha and Rodrigo Muniz scored in the space of four first-half minutes as the Cottagers took early control at Turf Moor, but the Clarets proved a different proposition in the second half.
Fofana, on loan from Chelsea, came off the bench to give the home side hope 19 minutes from time and then completed the fightback with 90 on the clock.
Bottom side Sheffield United showed no such battling spirit as they were thrashed 5-0 by Aston Villa at Bramall Lane.
The result was effectively settled in the first 20 minutes as John McGinn, Ollie Watkins and Leon Bailey all found the net for the visitors.
Youri Tielemans added a fourth with a fine effort on the half-hour and Alex Moreno completed the rout early in the second half, moving Villa back into the top four. The Blades could not even get a consolation as Vini Souza's late volley was ruled out for offside.
Crystal Palace also endured a miserable afternoon as they slumped to a 4-1 loss at rivals Brighton.
Lewis Dunk opened the scoring for the Seagulls after just three minutes before Jack Hinshelwood and Facundo Buonanotte struck in quick succession to make it 3-0 at half-time.
Jean-Philippe Mateta pulled one back for Palace, but Joao Pedro wrapped up victory for Roberto De Zerbi's side six minutes from time.
