Scarlett latched on to a through ball from Lucas Moura and slotted past the Orient goalkeeper in the first half.
But Orient equalised in the second half as new Tottenham head coach Nuno was held in his first game as head coach.
Eric Dier captained a Spurs side that included Joe Hart, Harry Winks, Lucas, Dele Alli and Steven Bergwijn.
There is real excitement around Scarlett and his finish showed why.
Mourinho gave Scarlett his Spurs debut last season, bringing him off the bench in two Europa League matches before making him Tottenham’s youngster-ever Premier League player when he brought him on against West Brom in February aged 16 years and 320 days.
Scarlett, who came through Spurs' academy, trained full-time with the Spurs first-team squad last season while he was playing for the Under-18s and development team.
Mourinho has predicted a bright future for the Londoner and, when speaking about talented player who could break into the England squad, said this week: “I know he’s still very young and it is probably too early for the next World Cup, but Tottenham have a striker in Dane Scarlett, who, if everything goes well, is going to be phenomenal.”
Scarlett signed his first professional contract with Tottenham in March after turned 17 to trigger a deal that runs until 2023.