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.
However, 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 goal-line clearance to keep out Dejan Kulusevski’s header shortly after the hour mark.
Postecoglou brought on former Forest winger Brennan Johnson and forward Dominic Solanke with a quarter of the match to go.
Matz 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.
Nuno said his team had toughed it out against the home side, who he admitted had his men “against the ropes”.
“It is about getting through it, especially in these moments of the season when there is fatigue, not only physical but mental, anxiety – all these things made the team go back a bit,” he told Sky Sports.
Nuno, whose side play Manchester City in the FA Cup semifinals this weekend, said his team had a big week ahead of them.
“You don’t get these moments often, especially in this league when we are fighting for something nobody expected,” he added.
Forest, who finished 17th last season, 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 semifinals of the Europa League next month, with the winner of the competition guaranteed a place in next season’s Champions League.
A disappointed Postecoglou said his side had dominated Monday’s game but had “conceded poor goals and given ourselves a mountain to climb”.
The Australian, whose team travel to face Liverpool on Monday, admitted the Europa League was now his priority as he tries to achieve momentum and protect his players.
Fifth-placed Manchester City host Villa on Wednesday.
