Former Wolves boss Nuno was not first choice to replace Jose Mourinho in the close-season and there have already been murmurings that Tottenham's players are bored of his training methods.
But Sunday's 2-1 home win against in-form Villa will buy the Portuguese boss some breathing space as players head into the international break.
Spurs looked shaky again when Ollie Watkins cancelled out Pierre-Emile Hojbjerg's first-half opener but a Matt Targett own goal in the 71st minute proved decisive.
The win lifts Spurs to eighth in the Premier League table, two points outside the top four.
"Good, good, good," Nuno told Sky Sports. "We needed a win but a well-deserved win. We did a good match against tough opponents.
"Important for us, the boys and the fans. We reacted well after conceding and positive in a lot of ways."
Leicester, who have grown used to life near the top of the table in recent seasons, travelled to Selhurst Park without a Premier League win since August.