The Frenchman put Brighton ahead from the penalty spot in the fourth minute when Allan Saint-Maximin brought down Tariq Lamptey after the teenage wing-back cut inside from the right and darted into the box.
Lamptey, 10 days shy of his 20th birthday, was also involved in Brighton's second three minutes later, releasing Leandro Trossard down the right.
The Belgian fired in a low cross that Maupay slotted home from close range and though the goal was initially disallowed for an offside in the build-up, referee Kevin Friend overturned his decision after consulting the video assistant referee (VAR).
The home side showed more fight after the restart and Callum Wilson spurned Newcastle's best chance to cut the deficit when he headed over from close range in the 63rd minute after Brighton were caught trying to play out from the back.
Trossard hit the post in the 71st minute before the visitors extended their lead late on when Aaron Connolly curled the ball home after being picked out by Maupay to complete a comfortable win for Brighton.
Newcastle, who opened their campaign with victory over West Ham United last weekend, struggled to contend with Brighton's 3-5-2 formation and failed to muster a shot on target.
"When you're 2-0 down after seven minutes then it becomes difficult against a team who base their style on possession," Newcastle manager Steve Bruce said.
"We've been beaten badly at home so we have to expect what's coming. You can't go up and down, which in the 15/16 months I've been here has happened. We've gone from having a decent week to throwing in a hand grenade like that so we have to accept the criticism."
The only blemish for Brighton, who lost their opening game of the season to Chelsea, was a red card for midfielder Yves Bissouma in the 89th minute for catching Jamal Lewis in the face with a wild swing of his boot.
Victory moved Brighton, who host Manchester United in the league next, up to eighth in the table while Newcastle slipped to 11th.