Son pounced on Alli's smart ball over the top to put Spurs in front after just 20 minutes at the Emirates.
Alli then coolly chipped it over keeper Petr Cech with the outside of his boot in the second half to make it 2-0.
The England midfielder was then hit on the head by a plastic bottle thrown from the crowd shortly after that goal.
It was a heated atmosphere at what was the second north London derby in 18 days; victory ensured Spurs made the last four for the first time since 2014-15.
Back then, they went on to the final and were beaten by Chelsea - their semi-final opponents this time around. Spurs will be at home in the first leg, scheduled for the week commencing 7 January, with the return at Stamford Bridge a fortnight later.
Holders Manchester City will take on League One side Burton in the other semi-final.
Tottenham keeper Paulo Gazzaniga made two good saves in the first half of Wednesday's quarter-final - denying Henrikh Mkhitaryan from close range and later tipping Aaron Ramsey's volley on to the post.
In the second half, Alexandre Lacazette hit the post and Pierre-Emerick Aubameyang's cross curled onto the top of the crossbar before Christian Eriksen's fizzing shot was saved by Cech.
Arsenal had not lost back-to-back matches prior to this defeat since they were beaten by Manchester City and Chelsea in the Premier League back in August.
Mauricio Pochettino's decision to make four changes and leave Harry Kane out of the starting line-up paid off. Son, Kane's replacement, caused problems in the first half and made his mark, scoring after just 20 minutes to make Tottenham's possession count.
The South Korean striker has now scored three goals in as many appearances in the League Cup this season.
When Kane was eventually introduced after 57 minutes - for his first appearance in the competition since September 2015 - he made an instant impact, setting up Alli with a perfectly weighted ball over the top.
The ability to rest and rotate will aid Tottenham in a busy festive period, which will see them play four matches in 10 days.