Manchester United staged a comeback to beat Brighton 2-1 and move four points ahead of Leicester in the race for second place in the Premier League.
He subsequently went off injured but Mason Greenwood headed in the winner seven minutes from time.
Joe Willock earned Newcastle a priceless Premier League point as they fought back to earn a 2-2 draw with Tottenham.
However, Steve Bruce's beleaguered side turned in a gritty display, which was rewarded when, seconds after Kane had hit the post, Arsenal loanee Willock smashed home an 85th-minute equaliser.
The result was another blow to Spurs' hopes of qualifying for the Champions League.
Aleksandar Mitrovic seized on a mistake by Tyrone Mings to put the Cottagers ahead in the 61st minute but Villa struck three times in the final quarter of an hour.
Substitute Trezeguet got the first two in the 78th and 81st minutes and Ollie Watkins built on his debut goal for England by grabbing the third as Villa ended a run of four games without a win.
Striker Ings, back after a month-long absence due to a leg issue, was involved in all three Saints goals to inspire the hosts to a first home league success in three months.
Nathan Redmond volleyed home the second-half winner after Ings had sparked the dramatic turnaround by setting up Stuart Armstrong's goal and then equalising before the break with a fine individual effort.