Antonio Conte's side went into the game knowing a draw would be enough to secure fourth spot given its far superior goal difference over fifth-placed Arsenal, but nothing less than a win looked likely after Dejan Kulusevski's early opener.
Spurs continued to dominate at Carrow Road after that and Harry Kane added a second before half-time after a howler from Tim Krul.
The impressive Kulusevski grabbed his second midway through the second half, before Son, who was chasing the golden boot, scored two excellent goals to move onto 23 goals in the top flight this season.
Kulusevski gave Spurs a deserved lead in the 16th minute, the Sweden international stroking in via a deflection off Jacob Lungi Sorensen after being teed up by Rodrigo Bentancur.
Spurs doubled the advantage in the 32nd minute when Kane headed in from eight yards from Bentancur's first-time cross after a dismal kick out by Krul.
The Norwich goalkeeper twice denied Son after the break, before Kulusevski saw a tame effort cleared off the line after he had rounded Krul.
The on-loan Juventus forward was not to be denied in the 64th minute, though, as he whipped into Krul's top-right corner from 15 yards.
Son stroked into Krul's bottom-left corner from Lucas Moura's clever pass in the 70th minute to make it four, with the South Korea international then curling in from 25 yards five minutes later to add further gloss to the scoreline.