It was Adebayors eight game with just one goal scored this season.
Spurs, on 20 points, go top of the league for the first time since August 2022 and remain unbeaten after eight games.
It is the North London clubs best ever start to a Premier League season.
For Adebayo and Luton, they are in 18th place on four points.
Spurs were reduced to 10 players after Yves Bissouma was sent off after receiving a second yellow card four minutes into first half added time.
But it was Spurs who took the lead in the 52nd minute thanks to Micky Van de Ven off and assist from James Maddison.
Born in Brent, England, Adebayo signed for Luton on 1 February 2021, for an undisclosed fee.
The 25-year-old scored on his first start in a 11 draw against Millwall on 23 February 2021.
Last season he scored eight goals in 45 games in the Championship for Luton.