Captaining the team on his 300th appearance in all competitions, Kane became the third Spurs player to score a double century of goals when he headed in Lucas Moura's corner in the 13th minute.
The England international and Lucas then reversed roles for the latter to double the scoring after the half-hour mark, with Jose Mourinho's decision to make seven changes from the side that lost 1-0 to Royal Antwerp in their previous Group J outing paying dividends.
Kane was withdrawn at half-time and Claudiu Keseru pulled one back for Ludogerts but Giovanni Lo Celso restored the visitor's two-goal cushion 12 minutes later and they cruised to the three points.
It seemed like Kane's wait for his landmark strike would not be ended in Bulgaria when he lofted a glorious one-on-one opening over and rattled the post with a powerful drive inside the opening six minutes.
The third time proved the charm, however, as his next effort was a powerful header from Lucas' corner that Plamen Iliev could not keep out.
Iliev denied Matt Doherty before Gareth Bale slid a pass into Kane, who returned the favour for Lucas with a perfectly weighted square ball that the Brazilian tapped home.
Seemingly sensing the job was done, Mourinho replaced Kane with Carlos Vinicius for the second half and Keseru halved the deficit in the 50th minute with a volley after Bernard Tekpetey's shot looped up off Harry Winks.
Son Heung-min replaced Lucas in the 61st minute and 15 seconds later teed up Lo Celso to ensure that Spurs' Europa League campaign was put back on track.