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The win lifted Forest to third place in the Premier League with 34 points, while Spurs remain in 11th with 23 points after 18 games.
In the 28th minute, Forest launched a counter-attack, with Morgan Gibbs-White threading a pass through the Tottenham defence for Elanga to score his third goal of the season.
Despite Spurs controlling possession, Forest maintained their defensive solidity, recording a second consecutive clean sheet.
Nuno Espirito Santo, managing against his former team, celebrated the crucial win.