Murillo's header had given the hosts a half time lead at the City Ground, but Alexander Isak's opportunistic leveller - his fourth goal in four games - drew the Magpies level ten minutes after the restart.
Joelinton's superb effort then put the visitors in front, before substitute Harvey Barnes made sure of the points inside the last ten minutes with a well-taken finish after a quick counter-attack.
The result lifts Eddie Howe's side to eighth place in the Premier League table - just a point shy of the Champions League places.
Former Magpie Chris Wood saw a shot blocked by Lewis Hall early on, while Joe Willock fired wide and Anthony Gordon blazed over, but Forest took the lead after 21 minutes after Willock had brought down Anthony Elanga on the right. The Swedish winger's free kick was met by centre back Murillo, who planted a firm downward header beyond Nick Pope to hand the hosts the advantage.
United looked to respond quickly, and Bruno Guimarães stung the palms of goalkeeper Matz Sels, another ex-Newcastle player, who pushed away the Brazilian's left-footed drive soon after Howe's side fell behind.
But while Forest were a constant threat on the break with the speed of Elanga and Callum Hudson-Odoi, United ended the half on the front foot, with Sels saving from Dan Burn after a neat exchange between Joelinton and Sean Longstaff on the edge of the box. They looked reinvigorated after the break too, with Guimarães narrowly off target in front of the travelling Newcastle supporters.
But after Pope had been forced into a save from Nicolás Domínguez after another Forest breakaway, United drew level. Gordon's corner struck Wood in the box and dropped kindly for Isak, who finished sharply from close range to level the scores.
Ryan Yates - who had earlier avoided punishment for a first half elbow on Tino Livramento - saw a header held by Pope but the hosts' threat seemed to dim a little after the withdrawal of Morgan Gibbs-White, with yet another former Magpie, Elliot Anderson, introduced to face his old club in league action for the first time. Howe's side looked the likelier to snatch a second, with Isak connecting superbly with Willock's cross to the back post with a quarter of the game remaining but seeing his volley flash just past the upright, with Sels well beaten.
But seconds later, United did register a second. It was a terrific finish from Joelinton, who picked up the ball on the right from Isak before driving forward and curling home with his unfavoured left foot, the ball clipping the post on its way in to put his side in front for the first time.
Forest rallied briefly, with Elanga failing to connect properly with a headed attempt in the box, but any hopes they had of snatching a point were extinguished on 82 minutes when Barnes stepped off the bench to roll home a third and seal the victory. It was a typical finish from the former Leicester City winger, who cut inside from the left to make space for a low shot which beat Sels at his near post, giving head coach Howe the perfect start to his fourth year in charge of the Magpies.
