Newcastle had looked like being made to settle for a draw but Isak's spot-kick penalty earned a deserved three points which sees them move to within a point of fourth-placed Tottenham with a game in hand.
Forest had gone ahead midway through the first half through Emmanuel Dennis' first goal for the club, but Isak levelled in superb fashion on the stroke of half-time when he volleyed home.
A second half of frustration unfolded for Eddie Howe's side as they searched for a winner, not least when they had a goal controversially ruled out by VAR.
But Isak had the final say from the spot after Moussa Niakhate's handball to keep the dream of Champions League football next season firmly alive.
The late goal saw Forest beaten at the City Ground for the first time in 10 matches going back to September and they could end the weekend's action back in the relegation zone.
Victory was only what Newcastle deserved as they did not start like a team who had yet to win away in 2023.
They had their hosts on the rack in the first 15 minutes and created two big opportunities to take an early lead.
They should definitely have gone ahead in the sixth minute as a nice move down the right saw Jacob Murphy put his cross into a dangerous area and Joe Willock got there first, but he fired his first-time effort well over under pressure from Serge Aurier.
Six minutes later they came within inches of scoring as a clever free-kick saw Kieran Trippier slip in Isak and the Swede's shot deflected off Aurier on to Renan Lodi and then on to the underside of the crossbar.
Forest weathered the storm and they took the lead in the 26th minute, though it came gift-wrapped from the visitors.
Sven Botman looked in control of Jonjo Shelvey's long ball, but he passed back to goalkeeper Nick Pope without looking and Dennis was there to intercept and dink home in style for his first goal for Forest.
The hosts were enjoying a spell of pressure but Newcastle again rattled the crossbar in the 35th minute.
Murphy stole possession off Lodi and teed up Longstaff on the edge of the area and his shot deflected off Felipe and onto the bar.
But finally they had some luck with the woodwork as they found the perfect time to level, with Isak making it 1-1 in first-half injury time.
With Forest appealing for a free-kick, Willock broke the lines down the right and sent in a cross that was slightly behind Isak, but the striker produced a fine volley that went in off the post.
It was a lively start to the second half and Newcastle were again the aggressors as half-time substitute Elliot Anderson almost scored with his first meaningful touch but Keylor Navas tipped over a vicious shot.
Navas was again called into action when he palmed over Bruno Guimaraes' fierce shot as the visiting pressure grew.
And Newcastle thought they had made it count in the 63rd minute as Anderson headed home Isak's cross at the far post in front of a packed away end that went wild in celebration.
But it was short-lived as after VAR intervention, Tierney ruled that Longstaff interfered with Lodi's clearance prior to the cross when he was in an offside position.
It was all Newcastle in the final 20 minutes as they searched for a winner and it eventually came in injury time as Niakhate's raised arm made contact with the ball and Isak coolly converted from the penalty spot.