The Reds had to come from behind twice as two excellent strikes from Ameobi cancelled out an early opener from Rhian Brewster and a penalty from Andre Ayew.
Sabri Lamouchi made just one change from the side that drew 1-1 at Preston North End on Saturday as Tobias Figueiredo came in to replace Michael Dawson while there was also a place on the bench for the returning Joe Lolley.
The visitors needed the three points and made much the brighter start on Trentside. Jake Bidwell's terrific left-wing cross with the outside of his boot was met at the back post by Ayew but the Ghanaian's powerful header was just off-target.
That early miss didn't deter Steve Cooper's side though and they took the lead after only eight minutes. Bidwell was again the provider and this time his low cross was swept home in outstanding fashion by Brewster as the Liverpool loanee netted his ninth goal of the season.
Forest were struggling to find any sort of fluidity or rhythm in their game but they equalised against the run of play on 20 minutes. Ameobi jinked past a couple of players and made his way to the edge of the Swansea penalty area before he curled a delightful effort beyond the dive of Erwin Mulder into the corner of the net.
Ameobi's goal seemed to breathe life into Forest's performance and they should have been ahead just before half-time. Samba Sow fed the ball through to Ameobi on the right-hand side, he powered into the penalty area but slotted his low shot just wide of Mulder's far post from close range.
That miss was to prove costly for The Reds as they fell behind in the final minute of normal time in the first half. Ayew was brought down by Yates, just inside the box according to referee Oliver Langford, and he stepped up himself to send Brice Samba the wrong way and put the visitors in front at the break.
Swansea were finding Ameobi a tricky customer to handle and he levelled up the contest once again ten minutes into the second half. Ameobi cut in from the right-hand channel as the Swansea defenders backed off and he unleashed an unstoppable 25-yard low drive into Mulder's net.
The second half was an entertaining yet attritional affair as both sides went for the three points they needed. Ayew had a golden chance to put Swansea back in front from six yards out but Samba was somehow able to make the save by kicking the ball back off Ayew and out for a goal kick.
Eight minutes of normal time were left on the clock when Naughton was sent off. Alex Mighten sprinted away down the left-hand side as Forest counter-attacked from a Swansea corner and Naughton brought him down with a dangerous lunge and Langford was straight over with the red card.
There were late opportunities on the edge of the Swansea penalty area for Carl Jenkinson and Lolley but Forest couldn't find that crucial third goal and must wait a few more days to secure their place in the Sky Bet Championship play-offs.
Forest: Samba; Jenkinson, Figueiredo (Dawson 76'), Worrall, Ribeiro; Watson (c); Da Costa (Lolley 76'), Sow (Silva 56'), Yates (Diakhaby 88'), Ameobi (Mighten 76'); Grabban
Subs not used: Smith, Bostock, Darikwa, Johnson
Goals: Ameobi 20', 55'
Bookings: Sow 32', Figueiredo 38', Jenkinson 49', Silva 84'
Swansea: Mulder; Naughton, van der Hoorn (Cabango 63'), Guehi; Bidwell, Grimes (c), Fulton (Celina 70'), Gallagher (Byers 86'), Roberts; Ayew, Brewster (Garrick 86')
Subs not used: Gould, Dyer, Routledge, Dhanda, Cullen
Goals: Brewster 8', Ayew (pen) 45'
Bookings: Gallagher 45', Byers 90'
Sent off: Naughton 82'
Referee: Oliver Langford
