Almirón's well-taken early opener set Eddie Howe's men on their way amid a raucous atmosphere before full back Burn thumped home a brilliant header to make it two before the break.
Longstaff then fired home to make it 3-0 before Lucas Hernández pulled one back, but two decades on from their last appearance in the competition, this was Newcastle's night and it was capped by Schär's magnificent swerving drive in stoppage time.
The result puts Howe's men top of Group F after matchday two, one point ahead of the Ligue 1 outfit, who were well beaten by the end of a breathless encounter.
After the teams had emerged at a deafening St. James', the French side fired an early warning - Kylian Mbappé lofting a pass over for Ousmane Dembélé, whose well-struck volley flashed just past the upright.
But United fed off the energy in the stadium and PSG looked overawed by the hosts' conviction in attack. Almirón blazed one over after a Achraf Hakimi error but on 17 minutes the Paraguayan wasn't so forgiving. This time it was Marquinhos making the mistake, chipping a pass out of his own box which Bruno Guimarães intercepted, heading towards Alexander Isak. The Swedish forward's low strike was brilliantly kept out by Gianluigi Donnarumma but the rebound fell straight to Almirón, whose firm effort found the corner and raised the roof.
Schär whipped a clever corner from Kieran Trippier just past the post soon after and apart from a deflected Goncalo Ramos attempt Luis Enrique's side struggled to threaten again in the first half. They were sluggish in the final third and five minutes before the break, they fell further behind. A Trippier set piece caused chaos in the box, with Donnarumma saving well again from his own man and then Guimarães, who tried to squeeze one in from a very acute angle. But the Brazilian's cross when the ball was worked back to him was perfect for someone to attack.
That someone was Burn, who steamed in and towered over Milan Skriniar, planting a firm header beyond PSG's Italian stopper who somehow managed to scramble the ball out and away. Replays showed the ball had comfortably crossed the line but play was halted by referee István Kovács, with an offside call seeming the most likely reason to those inside the ground. Confusion reigned for what seemed like an eternity with the prolonged VAR check presumably looking for an offside, handball or an incorrect call about whether the ball had gone in - but eventually the Romanian official blew his whistle and pointed to the centre, allowing local lad Burn to bask in the adulation and pandemonium that ensued.
It was much of the same after the restart. PSG were sloppy in possession, hassled and harried and given no breathing space by this tireless Magpies side. Gaps appeared at an even greater rate and five minute into the second period, Longstaff spotted one, charging into the box to pick up Trippier's neat ball infield. He struck it first time, low and across goal, and Donnarumma couldn't get down quick enough with the ball zipping under him and bouncing up into the roof of the net.
Newcastle were in dreamland but their joy was tempered ever so slightly five minutes later. Teenage midfielder Warren Zaire-Emery's delicate dink over the United backline was despatched by Hernández, who survived a brief VAR check to pull a scarcely-deserved goal back for Enrique's charges.
That goal from the full back preceded a brief spell of pressure from the visitors, with Dembélé racing clear before skewing his shot wide. The Frenchman then wriggled free of Burn and smashed one goalwards but it was beaten away by Nick Pope who, in truth, hadn't been required to make too many stops, such was the resilience of Newcastle's defence.
There was a snapshot from Mbappé and a wild attempt from Vitinha but neither troubled Pope and regardless, this was the Magpies' night. It was capped, in sensational fashion, by centre back Schär: 25 yards from goal, in stoppage time, he heeded the calls from the crowd and shot. It was perfect. Donnarumma's dive was despairing and the Gallowgate End rose in unison as the ball nestled in the top right-hand corner, curling in beautifully. For those who have waited 20 long years for Champions League football to return to Tyneside, it doesn't get much better than this.
