England striker Kane was injured after an innocuous-looking challenge with Jake Cooper in just the eighth minute at White Hart Lane, but it did little to stem Tottenham's blistering attack against the League One side.
Christian Eriksen, Kane's replacement, put the home side ahead after 31 minutes and Son's superb curling effort made it 2-0 before half-time.
Son delighted the home fans with another fine effort, watching a long ball over his shoulder and volleying home, before Dele Alli made it 4-0 and White Hart Lane was treated to Vincent Janssen's first goal from open play in his 30th appearance for the club.
The rout was complete by Son's first hat-trick for Spurs in injury-time to take his tally in this season's competition to six goals, the most of any Spurs player, and they can look forward to a last-four tie at Wembley.
Tottenham will now assess Kane, who will also be a concern for England ahead of a double header with Germany and Lithuania this month, but Pochettino will be delighted with how his Spurs attack stepped up in his absence on what is their last FA Cup game at White Hart Lane before a new stadium is built its place.
Spurs poured forward early and Harry Winks forced a save from Tom King with a curling shot just outside the area.
The hosts' flow was interrupted when Kane turned his ankle under a challenge from Jake Cooper and had to be replaced by Eriksen.
Alli saw strong penalty claims turned away when he was hauled down by Millwall captain Tony Craig, while at the other end Steve Morison arrowed a shot just over from 20 yards.
Spurs initially struggled for a focal point in Kane's absence, but they went in front in the 31st minute.
Alli chested the ball down in the area and Eriksen half-volleyed his shot low into the bottom-left corner, leaving King with no chance.
Spurs doubled the lead shortly before the break when Alli combined with Eric Dier to feed Son, who recovered from his initial poor control to turn his marker and bend a glorious left-foot shot past a helpless King.
The Millwall goalkeeper was called into action as soon as the second half began, saving Eriksen's long-range drive at full stretch but there was nothing he could do to stop Son making it 3-0 after 54 minutes.
Kieran Trippier sent a long ball into the box and it fell over Son's right shoulder before he hit it on the volley and under King's body.
Millwall still had not registered a shot on target when Trippier broke down the right flank and passed to Eriksen, whose cross found Alli unmarked at the far post and he scored his 16th goal of the season with a simple tap-in.
Janssen had only been on the pitch for five minutes when Son burst into the box and squared the ball to the striker, who slotted home a fifth.
King had been outstanding for Millwall and he was unfortunate to be beaten for a sixth time when Son met Eriksen's cross with a shot that squirmed under the goalkeeper's body and ensured the matchball left the field in his arms.
Key Opta stats:
- Spurs have progressed from 14 of their last 15 FA Cup ties against teams from a lower division.
- Spurs have won an FA Cup game by a six-goal margin for the first time since January 1973 vs Margate.
- Indeed, this is Millwall's first defeat in 18 matches in all competitions, having won 11 of the previous 17 (D6).
- Spurs have reached the FA Cup semi-final stage for the 20th time; the eight team to achieve this feat.
- Dele Alli scored in three consecutive appearances at White Hart Lane for the first time.
- Christian Eriksen has scored nine goals in all competitions this season, surpassing his tally from 2015-2016 (8).
- Eriksen has provided more assists in all competitions this season than any other Premier League player (16).
