David Wagner's side had the better of the first half in terms of possession and chances but still found themselves a goal down at the break.
Portuguese forward Hildeberto Pereira lofted home neatly after Nicklas Bendtner's effort rebounded into his path in the 25th-minute.
But two goals in seven minutes swung the match the home team's way - first Kasey Palmer's header from a corner rebounded down off the bar and went in off keeper Vladimir Stojkovic.
Then a teasing cross from Rajiv van La Parra was turned into his own net by Michael Mancienne.
To complete the Forest defender's misery he was sent off late on after a second bookable offence in front of a 22,100 gate including 1,541 visiting supporters.
1 min: Debutant Town keeper Joel Coleman gathers as Matty Cash lurks.
3 mins: Elias Kachunga's cross forces Forest defender Joe Worrall to chest the ball back to keeper Vladimir Stojkovic.
17 mins: Tommy Smith's curling cross ends up on roof of net.
21 mins: Ben Osborn free-kick stretches Town's defence, but the ball is cleared.
22 mins: Rajiv van La Parra curls a shot wide.
25 mins: Hildeberto Pereira lofts home for Forest after Nicklas Bendtner's shot ricochets into his path. 0-1.
28 mins: Nahki Wells curls a shot wide.
35 mins: Great interplay by Kasey Palmer and Kachunga, whose cross is hacked clear by Cash.
41 mins: Wells shoots over from distance.
42 mins: Kachunga's low shot is held.
45+1 mins: Aaron Mooy's effort is well blocked by Jack Hobbs.
53 mins: Town level from a corner as a Palmer header hits the bar and goes in off keeper Stojkovic. 1-1,
60 mins: An own goal by Michael Mancienne from van La Parra's cross puts Town ahead. 2-1.
69 mins: Town sub Jack Payne drills a free-kick into the Forest wall.
73 mins: Forest sub Britt Assombalonga shoots over.
76 mins: Mooy shoots over.
Town: Coleman, Smith, Hefele, Schindler, Holmes-Dennis (Lowe, 60), Hogg, Mooy, Kachunga, Palmer (Payne, 60), Van La Parra (Cranie, 88), Wells.
Subs not used: Billing, Bunn, Cranie, Stankovic, Coddington.
Forest: Stojkovic, Mills, Hobbs, Worrall (Carayol, 81), Mancienne, Vaughan, Cash, Osborn (Lam, 78), Lichaj, Pereira (Assombalonga, 63), Bendtner.
Subs not used: Dumitru-Cardoso, Grant, Vellios, Henderson.
Referee: Chris Kavanagh (Manchester).