The pair found the net for the Championship leaders in a victory over third-placed Forest which has left the club potentially just 90 minutes away from the Premier League.
If Chris Hughton's side - who were watched by jubilant owner Mike Ashley - win at Peterborough United on Saturday and Forest lose at Bristol City, they will be guaranteed a place in the top two of the Championship.
Ameobi, on his comeback from injury, struck 19 minute from time after coming on for the injured Andy Carroll early in the second half.
And Enrique made sure of the points in injury time with his first goal for the club in 88 appearances, with the strike sparking huge celebrations both on the pitch and the terraces.
Newcastle, with Peter Lovenkrands, Jose Enrique and Mike Williamson back in the starting XI, had the better of the first half but they couldn't conjure up a goal in front of an expectant crowd of 45,987.
Carroll, again wearing gloves to protect his bandaged hands, had United's best two chances before the break.
The big striker, who scored the winner for Newcastle at Doncaster Rovers last Tuesday a day after hitting the headlines over an alleged bust-up which left team-mate Steven Taylor with a broken jaw, saw a volley turned round the post by Lee Camp in the 12th minute.
And midway through the half the ball rolled to him after Lovenkrands had a penalty appeal waved away but Carroll chose not to shoot with his weaker right foot and he couldn't get a sight of goal once he had shifted the ball to his left.
Forest went on to keep United largely in check for the rest of the half, although they survived a late scare when Lovenkrands and captain Kevin Nolan were involved in a goalmouth scramble.
Carroll limped off in the 49th minute to be replaced by fit-again Ameobi. The eight-goal striker having been sidelined since early last month with a hamstring problem.
Billy Davies' men grew in strength as the half wore on but they couldn't trouble Steve Harper in the Newcastle goal.
Ameobi, however, made his first chance count with the 28-year-old spinning off James Perch after Nolan found him on the edge of the box to shoot low into the far corner of Camp's goal.
Newcastle, buoyed by their breakthrough, then seized control of the game.
Nolan forced a good save from Camp at his near post in the 79th minute and Jonas Gutierrez saw another effort deflected over the crossbar.
But it was the unlikely figure of Enrique who made sure of the points in injury time when the left-back beat Camp with his weaker right foot to leave his team on the brink of a remarkable return to the top-flight after just a season in English football's second tier.