Lucas Torreira was handed a straight red during added-on time as Spurs maintained their four-point advantage over Arsenal in third position.
Arsenal could have gone ahead inside the opening two minutes with Alex Iwobi's pass being deflected into the path of Alexandre Lacazette, but the Frenchman snatched at a half-volley on the turn with his shot going well wide of the far post.
Spurs were proving to be a match for their fierce rivals at the quarter-hour mark, but Arsenal went in front shortly afterwards after Kane had squandered a shooting chance at the other end.
A clearance was not dealt with by Davinson Sanchez on the halfway line with Lacazette being able to play Ramsey through on goal, and the Welshman held off pressure from Victor Wanyama to round Lloris and slot home into an empty net.
Kane thought that he had found an equaliser eight minutes later when powering a super header into the top corner, but the forward was one of three Spurs players who were half a yard offside from a free kick.
Iwobi came close to doubling Arsenal's lead as the first half approached a conclusion, but Lloris was equal to the Nigerian's curling effort from 15 yards and it acted as the catalyst for Spurs to end the opening 45 minutes on top.
However, Bernd Leno somehow kept his side ahead at the break, with the German stopper making a timely block to deny Christian Eriksen from close range before managing to react quickly in order to tip Moussa Sissoko's rebound over the bar.
Unai Emery opted to introduce Torreira at the break, and the fresh legs helped Arsenal continue to win the battle in midfield with Spurs struggling to create opportunities.
The visitors should have netted a second before the hour mark when Nacho Monreal pulled the ball back for Lacazette, but the frontman sliced his shot well wide of the far post from 10 yards out.
Toby Alderweireld then squandered a chance at the other end with the defender volleying wide at the back post, but it represented Spurs' best opportunity before the closing quarter of the game.
Arsenal looked comfortable but Spurs were gifted a way back into the contest with 17 minutes remaining after Shkodran Mustafi barged into Kane inside the penalty area.
Kane had been half a yard offside but it was missed by the assistant, leaving Spurs' top goalscorer to step up and send Leno the wrong way with a confidently-taken spot kick.
Both teams went in search of a decisive third and although chances were not being created, Arsenal were provided with the opportunity to score their own penalty when Sanchez was harshly adjudged to have pushed Aubameyang.
However, the Gabon international's effort was saved by Lloris down to his right, before Jan Vertonghen made a stunning block on the line to prevent Aubameyang from tapping home the rebound.
Torreira was dismissed in the final minute of added-on time for a late studs-up challenge on Danny Rose, but the sending-off mattered little in the context of the game as both clubs were forced to settle for a point.
TOTTENHAM HOTSPUR (3-4-3): Lloris; Alderweireld, Sanchez, Vertonghen; Trippier, Wanyama (Lamela 59'), Sissoko, Rose; Eriksen, Kane, Son (Llorente 79')
ARSENAL (4-2-3-1): Leno; Mustafi, Papastathopoulos, Koscielny, Monreal; Guendouzi (Torreira 46'), Xhaka; Mkhitaryan, Ramsey (Ozil 72'), Iwobi; Lacazette (Aubameyang 56')