The England striker, 24, netted early on after powering clear on goal then added a second when he sent a fine curling effort into the corner from the edge of the area.
Although there was to be no hat-trick for the England striker, he joined former Arsenal star Bergkamp as the only players to score twice in three successive away fixtures.
After another scoreless August, Kane has been oustanding in September, scoring 13 times for club and country, a figure unsurpassed in his career to date.
Highlighting just what a successful period it has been for Kane, he matched the highest tallies in a single month by Lionel Messi during March 2012 and Cristiano Ronaldo, who scored as many in October 2010.
13 - This equals the highest tallies in a single month by both Lionel Messi (13 in March 2012) & Cristiano Ronaldo (13 in Oct 2010). Peers. https://t.co/CJmXkdNoHk — OptaJoe (@OptaJoe) September 30, 2017