Just three points now separate the five clubs jostling for the three qualification slots still available but Forest, who ended a two-game losing streak, once more top that mini-league thanks to first-half goals from Elliot Anderson and Chris Wood.
Spurs, who saw Richarlison pull one back late on, could also be in the Champions League next term of course, by winning the Europa League. Astonishingly so given that this was their 18th Premier League defeat this season and they lie 16th in the table.
Postecoglou rested regular picks with next month's semi-final with Norwegian dark horses Bodo/Glimt in mind and although Spurs could claim to be superior overall and especially in the second half, the result won't have made the embattled Australian any new friends among an already-divided fanbase.
There is no doubt though that Forest fans love Nuno Espirito Santo, the man Spurs sacked after just four months in 2021 for the crime of being ninth.
And on Sunday last term's relegation escapees can enjoy another unexpected bonus in the shape of a Wembley FA Cup semi-final against Manchester City.
Forest had occupied third spot from December 29 until six days ago but wins for Newcastle, City and Chelsea saw them kicked down to sixth before kick-off.
The game's first chance saw Gugliemo Vicario bat away Morgan Gibbs-White's long-ranger - and Forest used the corner to take a fifth-minute lead.
It took two Spurs men to clear to where Anderson was lurking just outside the box and though hit powerfully his drive required a deflection off Rodrigo Bentancur to outfox Vicario with a bounce right in front of the goalkeeper.
Another shot went through the Italian five minutes later but a sliver of shoulder belonging to 'scorer' Wood was labelled offside by VAR.
Wood's reply was to nod home Anthony Elanga's cross in the 17th minute and again Vicario was culpable, by coming for a punch and failing to connect. It was the striker's first Forest goal in seven games and certainly one of the easiest of his 19 for them so far this season.
Spurs had chances to reduce the arrears but Mathys Tel blazed wide and Richarlison was off target with two headers. Inevitably the half-time whistle was met with home boos.
Postecoglou subbed centre-back pairing Micky van de Ven and Cristian Romero at the break, the latter so he couldn't be sent off having assaulted Anderson three times in the first half.
Gibbs-White, sent clear by Wood, slashed an opportunity for 3-0 wide before Harry Toffolo, making his first Premier League appearance of 2025, hooked away Dejan Kulusevski's glancing header from Pedro Porro's corner just as it was crossing the line.
Matz Sels made two good saves to deny Richarlison before the Brazilian finally found the net in the 87th minute by heading home Pedro Porro's cross. Forest held out however.
TOTTENHAM HOTSPUR (4-3-3): Vicario 5; Porro 6, Romero 5 (Danso 46, 6), Van de Ven 6 (Davies 46, 6), Spence 6; Kulusevski 6 (Solanke 67, 5), Bentancur 6, Sarr 6 (Bergvall 81); Odobert 5 (Johnson 67, 4), Richarlison 7, Tel 5.
NOTTINGHAM FOREST (4-3-3): Sels 7; Williams 7, Murillo 7, Milenkovic 7, Toffolo 7; Dominguez 6 (Yates 58, 5), Anderson 7, Danilo 6 (Hudson-Odoi 58, 5); Elanga 6 (Morato 46. 5), Wood 7 (Awoniyi, 75), Gibbs-White 7 (Sosa 90).
Referee: Peter Bankes 6