Fellow London side Brentford, playing in their first semi-final in a major cup competition, started brightly but fell behind in the 12th minute when unmarked French midfielder Sissoko headed in a fine Sergio Reguilon cross from the left.
The visitors, who had beaten four top-flight teams on their way to the semis, thought they had drawn level after the break when striker Ivan Toney headed home but the goal was disallowed for a very tight offside, which was spotted by VAR.
Toney’s grounded knee was ruled to be offside before he rose to head in from close range after Ethan Pinnock nodded a cross back across the face of goal.
Jose Mourinho’s Spurs side responded to that let-off with South Korean forward Son collecting a fine pass from Tanguy Ndombele and bursting goalwards before slotting past goalkeeper David Raya to make it 2-0 after 70 minutes.
Brentford’s disappointment was compounded when Josh Dasilva was sent off six minutes from time for an over-the-top challenge on Tottenham’s Denmark midfielder Pierre-Emile Hojbjerg, who was immediately substituted with blood streaming down his shin.
The showpiece match will be held at Wembley on April 25, after the Football League moved the date back from February to increase the chances of fans being able to attend.
United host City in the other semi-final at Old Trafford tonight.