The German centre half's unlikely brace proved timely for Eddie Howe's side, with his first arriving inside the opening minute on the Magpies' first-ever trip to the Hill Dickinson Stadium.
Lewis Miley made it two with a shot that somehow deceived former Sunderland stopper Jordan Pickford, before Nick Woltemade's beautifully-executed lob gave United a three-goal half time lead.
Thiaw's second header of the contest made it four after the restart and though Kiernan Deswbury-Hall pulled one back with a neat finish, this was a dominant display and important win for Newcastle, who eased to their first league triumph on the road this term.
Howe made six changes to his side following the Magpies' midweek Champions League defeat in Marseille, the most notable of which saw Aaron Ramsdale come in for the injured Nick Pope to make his first Premier League start since his summer loan move.
But it was one of the starters from Tuesday night's loss in France who put the visitors in front inside a minute at the Toffees' new home. Anthony Elanga forced Pickford to turn his early effort past the post and from Miley's inswinging corner, Thiaw glanced home his first goal for the club.
That opener was the fastest in the Premier League so far this season - beating Phil Foden's strike against Leeds United earlier in the day by four seconds - and United's fifth fastest strike of the Premier League era. They looked like adding to it immediately, with Harvey Barnes forcing another save from Pickford, who was to endure a miserable 90 minutes.
But Everton looked dangerous after going a goal down, and looked to get in behind the Newcastle backline. Iliman Ndiaye and Jack Grealish probed to no avail, though after an expert James Tarkowski knockdown his centre back partner Michael Keane went close with a deflected drive. From the resulting corner, Thierno Barry looped a header over Ramsdale's crossbar.
Dan Burn did well to prevent a Dewsbury-Hall cutback on another of the hosts' forays beyond United's defence but the second goal stymied any momentum the hosts were building. Elanga beat Vitaliy Mykolenko on the right and though his cross was partially cleared, it led to a few seconds of chaos that Everton couldn't deal with. Tino Livramento's shot was heading off target until Burn instinctively threw a leg out to turn it back into Miley's path. His left footed shot was straight at Pickford, but the England stopper inexplicably allowed it to bounce between his legs and into the net.
It was something of a gift and it could have been three soon after when Lewis Hall - starting in the league for the first time since September - picked up a Livramento pass and saw his strike clip a heel and then Pickford's bar.
But just before the break, Howe's men did move further in front. It was a swift and efficient counter, starting with Ramsdale's throw to Livramento. Elanga scampered on to his pass down the right and squared for Woltemade, who had a remarkable amount of space and time to delicately lift the ball over Pickford and make it three.
David Moyes introduced Charly Alcaraz at half time in a bid to turn the tide that had swamped the Everton midfield. They pushed briefly, with Ndiaye forcing a straightforward stop from Ramsdale before a Grealish corner almost squeezed in at the front post until Barnes' intervention. The ball fell for the oncoming Alcaraz, though, whose effort thumped the post - and carried so much power that the rebound landed around 30 yards from the United goal.
It was only a brief threat in the end, with Barnes going close soon after. Woltemade turned neatly six yards out and looked certain to score, but Tarkowski blocked it on the goalline, with Pickford beaten. Just seconds later, and three minutes before the hour, they did score again. Hall's cross from the left was perfect for Thiaw, who rose to nod home his second of the evening.
Everton thought they had one back when the isolated Barry was picked out by Grealish before slicing into the far corner, but a swift VAR check saw it disallowed for an accidental handball as the frontman brought the cross down.
The consolation the Toffees did score was one of real quality. Tarkowski's long ball looked a hopeful one but Dewsbury-Hall's stunning first touch on the turn turned it into a big chance - which the midfielder duly took, slotting beyond Ramsdale with 20 minutes remaining.
Substitute Jacob Ramsey had a sniff of a fifth late on, with Pickford saving with his legs as he narrowed the angle. It would have added extra gloss to an already impressive performance from Howe's charges, who were as slick as the Toffees were lacklustre.
