Spurs' 3-0 north London derby victory over the Gunners on Friday (AEST) left just a one-point gap between the two rivals in the table and, while Antonio Conte's men disappointed during the visit of Burnley, Kane secured the desired result.
The relegation-battling Clarets initially did a fine job of keeping Spurs at bay until conceding what they felt was a harsh penalty, with Kane converting his 23rd successive spot-kick for Tottenham.
Burnley then enjoyed encouraging spells after the interval, with Ashley Barnes hitting the post, but Spurs – who saw Son Heung-min denied twice by Nick Pope – managed to cling onto another important win.
A dominant Spurs start did not yield many clear-cut chances, with Kane and Emerson Royal going closest when forcing straightforward saves from Pope.
Burnley grew as an attacking presence itself and should have taken the lead when Maxwel Cornet shot straight at Hugo Lloris in a one-on-one situation.
Kane fired wide as Spurs looked to be heading into the break frustrated, but a VAR review subsequently spotted a handball by Barnes in the build-up to that chance, and Spurs' talismanic striker was clinical from 12 yards.
Spurs' start to the second half was sloppy, however, and Burnley almost capitalised.
Josh Brownhill headed agonisingly wide, before Barnes smashed against the post from 25 yards just past the hour.
Son twice looked destined to finish Burnley off in the final 25 minutes, but Pope produced a fine save down low and then made a reaction stop to deny him late on – not that it mattered for Spurs.