The Premier League side broke the deadlock after the break when Maxim Dekker fouled Lucas Bergvall in the area and Richarlison chipped his penalty into the net in the 53rd minute.
Alkmaar's David Moller Wolfe received a second yellow card and was sent off in the 85th minute, dashing any remaining hopes for the Dutch side as Spurs maintained their perfect start to the competition.
Spurs forward Son Heung-Min sat out because he was "a bit sore" after Saturday's 4-1 win over West Ham in the Premier League, manager Ange Postecoglou said.
Tottenham have nine points, while Alkmaar have three.
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