After being frustrated for much of Sunday’s (AEDT) match at Tottenham Hotspur Stadium, Spurs took the lead when Kane converted a 59th-minute penalty, which he won after a foul on Jordan Pickford.
Before that, Everton had missed the best moments of the game, with Amadou Onana and Demarai Gray firing in the first half.
Any hope for Frank Lampard’s side was dashed when Hojbjerg’s deflected shot ended the contest.
A completely dominant start for Spurs produced just one opportunity, with former Everton talisman Richarlison heading in at the back post.
However, it was Everton who had the golden chance in the 23rd minute, Gray deflecting off Rodrigo Bentancourt, only to shoot one-on-one with Hugo Lloris.
Annan had a similarly glorious opportunity and, like his team-mate, was unable to produce a clear finish after taking advantage of some slack Spurs defending before Richarlison pushed the other way.
After Richarlison’s meeting with his old team was cut short by a shin injury shortly after the break, Spurs produced a clean sheet as Pickford superbly kept out Kane’s volley.
However, Pickford was at fault when Spurs finally took the lead, with the England keeper adjudged guilty of bringing down Kane after spilling Matt Doherty’s ball. The Tottenham striker made no mistake from 12 yards.
Kane almost made it two for Tottenham with a brilliant turn and strike, although Pickford reacted sharply.
The introduction of Dominic Calvert-Lewin was a much-needed goal for Everton, though Spurs grabbed the points in the 86th minute when Hojbjerg headed into the right-hand corner from 15 yards.