The untidy 41st-minute effort from close range nudged Ange Postecoglou's team ahead in the closing stages of a first half in which both sides enjoyed encouraging spells of attacking play.
The north Londoners then successfully resisted United's pursuit of an equaliser after half-time to ensure it was they who lifted the trophy in Bilbao and secured a place in next season's Champions League.
FIRST HALF - JOHNSON EDGES SPURS AHEAD
There was an incredible noise reverberating around the acoustically friendly San Mames Stadium as Spurs got the ball rolling, kicking towards their Lilywhite-dressed supporters, while United attacked in the direction of our own red wall.