Son settled a hard-fought clash at the Tottenham Hotspur Stadium when his free-kick caught Watford keeper Daniel Bachmann out of position in the first half.
After their protracted search for a manager to succeed to Jose Mourinho dragged on throughout the close-season, Tottenham were in danger of becoming a laughing stock.
Nuno was far from their first choice, but the former Wolves boss has hit the ground running north London.
Three consecutive 1-0 wins, including one against Premier League champions Manchester City, as well as progress in the Europa Conference League, suggest Tottenham could enjoy a better season than expected.
Nuno is the first Tottenham manager since Arthur Rowe in 1949 to win his first three league games in charge.
"We knew the threat of Watford, we managed it well. I'm very happy because the players worked very hard," Nuno said.
Watford midfielder Moussa Sissoko was back at Tottenham, just two days after leaving for the Hornets and he nearly laid on an early goal.
Sissoko picked out Juraj Kucka and his shot was headed off the line by Eric Dier.
Tottenham took the lead in the 42nd minute when Son swung in a free-kick from the left wing and Bachmann was caught out as it sailed past him into the far corner.
It was Son's second goal of the season after the forward's winner against Manchester City on the opening weekend.
At Molineux, Manchester United set an English top-flight record for unbeaten away league games in their first match since agreeing a blockbuster deal to sign Cristiano Ronaldo from Juventus.
Portugal star Ronaldo, who will return to United 12 years after leaving to join Real Madrid, was not available to make his second club debut this weekend as he waits to officially complete the move.
While Ronaldo waits in the wings, unbeaten United had Greenwood to thank for their second win in three matches.
As United fans waved Portugal flags and chanted "Viva Ronaldo", 19-year-old forward Greenwood scored for the third successive game.
United have now gone 28 successive away league games without defeat, passing the mark they shared with Arsenal.
"It's a special goal. Of course the keeper maybe could have saved it but he hit it so quickly," said United manager Ole Gunnar Solskjaer, who confirmed Ronaldo is expected to feature against Newcastle on 11 September.
"Both sides could have won it, of course we were under pressure for a spell but we had most of the possession."
Solskjaer's team were indebted to David De Gea for a superb double save that kept the score level before Greenwood's winner.
De Gea blocked Romain Saiss's close-range header from a corner, then somehow spread himself to thwart the defender's close-range follow-up.
Greenwood made Wolves pay for that miss in the 80th minute when he took Raphael Varane's pass and hit a low shot that flashed under keeper Jose Sa.
Wolves claimed Paul Pogba had fouled Ruben Neves in the build-up to the goal, but referee Mike Dean and VAR ignored their protests.
United climbed to third place, while Wolves remained pointless and goalless under new boss Bruno Lage.