Sherwood had declined to comment on his future in the hours before kick-off at White Hart Lane after a report claimed he would be dismissed at the end of the season.
The former Tottenham midfielder was only appointed to replace the sacked Andre Villas-Boas in December, but his position has come under increasing scrutiny in recent weeks following a disappointing run of results.
Holland coach Louis van Gaal, Southampton manager Mauricio Pochettino and Ajax boss Frank de Boer have all been strongly linked with the Tottenham job.
Even this convincing win is unlikely to save Sherwood, but he looks set to go out fighting and fittingly it was a goal from England Under-21 striker Kane that set the tone.
Lee Cattermole had given bottom of the table Sunderland a shock early lead.
But Emmanuel Adebayor equalised and Kane, one of the promising prodigies Sherwood championed during his time as the club’s director of youth development, marked his first top-flight start by putting the hosts ahead after the break before late goals from Christian Eriksen, Adebayor and Gylfi Sigurdsson sealed the demolition.
Tottenham climb back up to sixth place with the victory, while Sunderland lie seven points from safety and even their two games in hand may not be enough to save them.
“I can’t talk about anything about my employment, you will have to ask my employers about that,” Sherwood told Sky Sports after the final whistle.
“I don’t know if the talk does anyone any favours but we’ve just got to get on with it. All I was focused on was to win this game.
“I don’t want anyone to feel sorry for me. It’s been a great opportunity to manage this club and my record in the Premier League is second to none compared to all the other managers who have been here.”
Sherwood’s players initially looked distracted by all the speculation about their manager’s future and they fell behind in calamitous fashion in the 17th minute.
Wes Brown had already headed just wide moments earlier when Tottenham got themselves into trouble from their own throw-in.
After picking up a pass from goalkeeper Hugo Lloris, Vlad Chiriches played a horrendous square pass well away from any of his team-mates and Cattermole gratefully accepted the gift to slot into the unguarded net from 25 yards.
Togo striker Adebayor led the Tottenham response and curled an effort over the bar before equalising in the 28th minute.
Eriksen drilled a low delivery across the face of goal and when Brown failed to get a clearing touch, Adebayor was on hand at close-range to deflect the ball into the net off his chest.
Sherwood was keeping his cool on the touchline, but he finally showed the emotion that has made him such a volatile figure when Kane was denied a penalty following Carlos Cuellar’s challenge early in the second half.
The Spurs manager was on his feet again soon after, but this time it was in celebration as Kane put the north Londoners ahead in the 59th minute.
Once again Denmark playmaker Eriksen was the catalyst as he whipped a teasing cross in towards Kane, who timed his run well and stretched out a boot to poke the ball beyond Vito Mannone.
Sunderland boss Gus Poyet sent on Craig Gardner and Ignacio Scocco for Cuellar and Cattermole in a bid to break Tottenham’s dominance.
But man of the match Eriksen killed off the Black Cats in the 78th minute when he got the goal his superb display deserved, firing home via a slight deflection off Phil Bardsley from outside the area.
With lacklustre Sunderland waving the white flag in surrender, Adebayor tapped in from Kane’s pass in the 86th minute to make it four and there was still time for Sigurdsson to lash home in stoppage-time.