Martial's introduction for Marcus Rashford after 70 minutes was not well received by some Manchester United supporters - but manager Jose Mourinho's move proved to be a masterstroke as the French striker raced on to Romelu Lukaku's header to beat Spurs keeper Hugo Lloris with only nine minutes left.
It was a narrow victory United just about deserved after creating the better chances in the second half, Lukaku producing an outstanding save from Lloris and hitting the woodwork with a header.
Spurs missed the goalscoring threat and presence of the injured Harry Kane, creating little other than a chance which Dele Alli steered just wide from Christian Eriksen's cross with the game still goalless.
In the end, Mourinho's change and Martial's penalty box finish made the difference as United recovered from last week's surprise loss at Huddersfield Town.
When Mourinho removed Rashford and sent on Martial with 20 minutes left, sections of Old Trafford signalled - and indeed gestured - their disapproval at the move.
It was actually the catalyst for United's match-winning surge as they finally pierced the defensive barrier erected by Spurs manager Mauricio Pochettino.
Romelu Lukaku, who had endured a frustrating afternoon, burst into life with a run and shot that was brilliantly saved by Spurs keeper Hugo Lloris then had a header that bounced to safety off the post.
United built momentum after Mourinho's change and it was Lukaku who rose to send a header into Martial's path for a finish that was not clean but good enough.
Mourinho had criticism to contend with for his approach to the goalless draw at Liverpool a fortnight ago - but here he made the tactical tweak that proved decisive.
Pochettino was, with very good reason, outraged by Manchester City manager Pep Guardiola's description of Tottenham as "the Harry Kane team."
There is so much more to Spurs than that in the shape of Christian Eriksen, Dele Alli and a superbly-drilled defence - but here they missed that edge of menace the 24-year-old would have provided.
Spurs are not 'the Harry Kane team' in the slightest - but any side in the world would miss a striker in the sort of form England's captain is currently in.
Son Heung-min battled hard before he was replaced by Fernando Llorente but neither could replicate the sort of threat Kane has been producing in recent weeks.
Mourinho will also point out he was missing key midfielder Paul Pogba - but there was no doubt, despite the defiant message from Pochettino before the kick-off, that Spurs were dealt a huge blow to their plans by Kane's hamstring injury.
Lukaku was given a real show of faith by Mourinho when he persisted with the striker after deciding to make changes.
It could just as easily have been Lukaku who was withdrawn rather than Rashford as the game entered its final phase - but Mourinho stuck with his summer purchase from Everton and it paid dividends.
The £90m striker had been a figure of frustration for much of the game as he gestured to his team-mates for greater and quicker support as he battled against Eric Dier, Jan Vertonghen and Toby Alderweireld.
And yet, through sheer persistence and determination, he was able to make a key contribution to set up Martial's late winner.
Another key figure for United was the rejuvenated Ashley Young, employed down the left-flank, defending tirelessly and occasionally delivering the sort of dangerous crosses that were his trademark as a flying winger.
He fully deserved the standing ovation he received from Old Trafford when he was taken off in stoppage time.AGENCIES