Daryl Murphy’s 25th League goal of the season - the first Blues to reach that number since Ray Crawford in 1963 - gave Mick McCarthy’s side the lead in the first-half.
Forest equalised after the break through a Christophe Berra own goal but Sears sent Portman Road into dreamland with priceless winner late on.
A point at Blackburn next weekend gives Blues a place in the top six.
Town made one change from the side that drew at Wolves last weekend, Stephen Hunt in for illness victim Jay Tabb.
Blues had a scare in the first minute when Henri Lansbury was presented with a chance inside the box but his shot lacked power and Bart Bialkowski gathered comfortably.
Murphy headed over at the other end, Ben Osborn tested Bialkowski with a terrific strike from 30 yards, Michael Mancienne then heading clear as Murphy looked to get on the end of a Tyrone Mings cross.
Freddie Sears let fly from distance but his shot failed to trouble Karl Darlow in the Forest goal in an even start to the game but it was Blues who edged in front. As the Town fans paid tribute to popular fan Chris Reynolds, 21, who passed away recently with a minute’s applause on 21 minutes, Murphy smashed in a half volley from just inside to box that gave Darlow no chance.
Tommy Smith’s alertness cut out the danger from Dexter Blackstock as the Forest striker looked to get in on a ball over the top, Cole Skuse volleying over from the edge of the visitors’ box.
Smith headed over from Hunt’s free-kick as Town threatened a second, Mings getting behind the defence but seeing his cross hacked clear, Luke Chambers’ shot on the follow up blocked by Blackstock.
Darlow saved low to his right after Murphy had cut inside two defenders and let fly from the edge of the box but it needed a fingertip save from Bialkowski to deny Lansbury and keep Blues ahead at the break.
Murphy almost bundled his way through the Forest defence as the second-half got underway, Sears whipping in a cross that went wide of the far post. Smith then going down under a challenge in the box but referee Graham Salisbury didn’t give it a second glance.
Hunt sent a free-kick just over after he had been fouled 30 yards out but Forest were level on 53 minutes, Bialkowski beat out Chuba Akpom’s effort but Osborn fed the ball back into the box and Berra diverted it into his own net.
Hunt made a brilliant tackle after Antonio’s mis-kick had fallen to the feet of Akpom six yards out, Blackstock then denied by the flying Bialkowski as Forest enjoyed their best spell of the first hour.
Luke Varney replaced Hunt in Town’s first change and the Blackburn loanee flashed a header just wide of the post, Kevin Bru entering the fray for the last 20 minutes, Jonathan Parr making way.
Darlow prevented Sears getting on the end of Teddy Bishop’s cross with a good take, Murphy testing the Forest ‘keeper again with a low shot on the turn.
Bru’s cross was too heavy after Cole Skuse won possession on the edge of the Forest box, Town then making a final change with former Forest winger Paul Anderson on for Bishop.
Varney was inches away from edging Town in front, his header from Anderson’s cross beating Darlow but landing on the roof of the net.
Seven minutes from time, bedlam - Sears drifted past his marker and his shot took a big deflection before finding the net off the inside of the post to leave Town on the brink of the Play-Offs for the first time in 10 years, with Blackburn to come next weekend.
Town: Bialkowski, Chambers, Smith, Berra, Mings, Parr (Bru 68), Skuse, Bishop (Anderson 78), S.Hunt (Varney 64), Sears, Murphy. Subs not used: Gerken, Hewitt, Fryers, Ambrose. Booked: Anderson.
Forest: Darlow, Lichaj (Fox 61), Mancienne, Lascelles, Hobbs, Vaughan, Osborn (Walker 85) , Lansbury, Antonio, Blackstock, Akpom (Burke 78). Subs not used: Evtimov, Kane, Paterson, Grant.
Att: 25, 199