Eddie Howe’s side started brightly but Alli, who appears to be flourishing under Mourinho, was the catalyst for Spurs’ victory.
Toby Alderweireld’s raking pass created Alli’s opening goal, with the England midfielder racing on to Son Heung-min’s touch to tap the ball past Cherries goalkeeper Aaron Ramsdale.
Alli’s second came from the same source, as he superbly controlled Alderweireld’s pass before holding off Jack Stacey and flicking the ball over Ramsdale.
When Moussa Sissoko added Spurs’ third goal – with a close-range volley from Son’s exemplary cross – the hosts appeared to be en route to an emphatic win.
However, Harry Wilson’s introduction sparked the visitors into life, pulling one back with a superb free-kick.
The on-loan Liverpool midfielder reduced the arrears further six minutes into stoppage time, although it proved too little too late for the Cherries, who have now conceded 16 goals on their last five visits to Tottenham.
While smiles were in short supply towards the end of Jose Mourinho’s tenure at Manchester United, the Portuguese manager has started life at Tottenham with a cheerful and airy disposition.
With his proclamations of humility, celebrations with ball boys and decisions to deploy players into positions they favour, his approach has clearly had the desired effect.
His new employers have so far reaped the rewards, and this was the first time Spurs have won three consecutive matches (across all competitions) since April.
While this was far from a vintage performance overall, one player who appears to have benefitted under the new regime is Alli.
This game suggested he is not only back to his influential best as a playmaker behind Harry Kane and Son, but also as someone capable of making a considerable contribution in front of goal.
His first goal on Saturday was all about anticipation and timing, running adjacent to Son and ensuring he was on hand to deliver the simplest of finishes.
His second showcased all of his talents – running power, control, strength and the composure in front of goal to apply the deftest of touches under pressure.
While Alli was unable to register his first hat-trick for the club, it was still his first two-goal haul in a home game for Tottenham since January 2017 and ensured he received a standing ovation when he was withdrawn late on.
