But Southampton refused to lie down with their new-found confidence and grabbed a deserved leveller through Boufal with two minutes to go.
Both Jose Mourinho and Ralph Hasenhuttl played close to first-choice XIs and the raucous crowd were rewarded with a thrilling FA Cup clash.
Son had a decent chance to open the scoring nine minutes in as Lucas Moura and Dele Alli combined to release the South Korean down the left side of the box, but he fired just wide.
And they had a goal ruled out just before the break as Giovani Lo Celso found the net via a deflection off an offside Son.
Spurs put together the move of the first half as Alli raced away on the counter before feeding Moura with a perfectly-timed pass.
The Brazilian cleverly cut across Kevin Danso and looked set to break the deadlock – but his goalbound shot was turned wide by Angus Gunn’s foot.
Danny Ings showed what a clever player he is soon after as he got in front of Hugo Lloris to chest down James Ward-Prowse’s quick free-kick, before volleying goalwards.
But the England hopeful was denied by a goalline clearance by Japhet Tanganga.
Hasenhuttl’s side were dealt a blow when Stuart Armstrong came off with a suspected hamstring injury just before the break.
Ings has been in lethal form this term but his radar was just off here as he fired a Michael Obafemi cross agonisingly wide after the break.
Lo Celso then kept up his hot form to set up Son’s opener, gliding past four Saints players from his own half before finding the South Korean via Erik Lamela.
Son did the rest, finding the corner brilliantly with his 12th goal of the season.
Lo Celso and Son then combined again as the Argentine floated a ball over the top, but his team-mate could only volley over.
Saints have become a different side since their festive feast of positive results – with October’s 9-0 home defeat to Leicester feeling like forever ago now.
Hasenhuttl’s men pushed for a leveller and should have had one when a free Ings headed wide with five minutes to go.
But the hosts did not give up and got their rewards at the death as Ings squared to Boufal, who blasted past Hugo Lloris from inside the box.