Forest came into the game at the Tottenham Hotspur Stadium in sixth spot in the table after losses to Aston Villa and Everton.
But early goals from Elliot Anderson and top-scorer Chris Wood settled their nerves and they held on tenaciously despite a late goal from Richarlison to climb to third in the table.
Leaders Liverpool have one hand on the Premier League title and Arsenal are likely to finish second.
But behind them the competition is fierce for the top five places that will guarantee qualification for the Champions League next season.
Just three points separate Forest from Aston Villa in seventh spot, with five games remaining for all the contenders.
Nuno Espirito Santo’s visitors took the lead in the fifth minute following a corner when Anderson’s deflected shot beat Guglielmo Vicario.
Wood had the ball in the net five minutes later but it was ruled out by VAR for offside.
But the New Zealander doubled Forest’s lead in the 16th minute, nodding a cross from Anthony Elanga past Vicario.
Spurs dominated possession and had more shots on goal but provided little threat.
Forest had a chance to make it 3-0 10 minutes into the second half when Wood cushioned the ball into the path of Morgan Gibbs-White but the midfielder skewed his shot wide.
Forest defender Harry Toffolo produced a spectacular overhead goalline clearance to keep out Dejan Kulusevski’s header shortly after the hour mark.
Substitute Spurs defender Kevin Danso let fly from range but it did not trouble Matz Sels.
Postecoglou brought on former Forest winger Brennan Johnson and forward Dominic Solanke with a quarter of the match to go.
Sels produced a superb fingertip save low down to his right from a Richarlison header with about 10 minutes to go as Tottenham upped the pressure.
The Brazilian did get on the scoresheet in the 87th minute, powering a header home from a Pedro Porro cross but the visitors held on.
Forest are two-time European champions but have not played in the continent’s top club competition since the 1980/81 season.
Spurs’ 18th defeat of a miserable Premier League season leaves them just two places above the relegation zone.
But Postecoglou still has a chance to win silverware in his second season at the club.
Spurs face Bodo/Glimt in the semi-finals of the Europa League next month, with the winner of the competition guaranteed a place in next season’s Champions League. (AFP)
