It was United's biggest home defeat in the league since Manchester City won 6-1 in 2011.
The hosts were awarded a penalty within the first minute after Davinson Sanchez brought down Martial and Fernandes converted from the spot.
Son put Spurs ahead three minutes while United were reduced to 10 men after Martial slapped Erik Lamela after a confrontation in the penalty box.
Kane scored two minutes after Martial's sending off while Son notched his second on the stroke of half time.
"Manchester United doesn't lose many matches, to lose by six is of course not every day," Spurs boss Jose Mourinho said after the match. "Historical victory but in the end three points and three very important points because we lost five points at home.
"We prepared ourselves very well tactically and also psychologically, the team was ready. So ready that a penalty in the first minute and going 1-0 down did not affect the team at all.
"Some people can say you play against 10 men for a long time which is true but for me it is also true that 11 v 11 we were playing extremely well, we were creating huge problems, we were winning 2-1 and they could not find solutions to control us."
Shaw told the BBC: "It was a lack of concentration and then a mistake, another mistake, another mistake. We switched off. It really hurts.
"Manchester United shouldn't lose in this manner. It really hurts, it's embarrassing and we've let a lot of people down. We've let ourselves down and the staff down. We need to look in the mirror, we are nowhere near it at the moment. Maybe it's lucky that we've got a break now.
"When we went down to 10 men maybe we gave up, there wasn't that character on the pitch. We were too quiet and naive. There was many mistakes, we need to do more.
"It's embarrassing. I was embarrassed on the pitch. We need a long hard look at ourselves."