Spurs dominated the first half but couldn't make the most of their 12 shots on goal, although Moore and Pape Sarr brought good saves out of Elfsborg goalkeeper Isak Pettersson.
Jalal Abdullai nearly gave the visitors a shock lead early in the second half when his bicycle kick narrowly fizzed over the bar as the game became more open.
Lucas Bergvall should have buried a header just before the hour mark but could only find the grateful Pettersson in goal and then Tottenham's injury woes deepened when second-half substitute Radu Dragusin was forced from the field after an innocuous-looking incident.
Just when it looked like it would be one of those days, substitute Scarlett, 20, lit up Tottenham Hotspur Stadium when he buried Dejan Kulusevski's pinpoint cross in the 70th minute.
Fellow academy product Ajayi, 19, finished well on his Spurs debut with six minutes remaining and youth player Moore, 17, wrapped up the win with a well-taken goal in second-half stoppage time. The 3-0 victory meant Tottenham finished fourth in the Europa League table to qualify for the last 16 of the competition.