However, Chelsea threatened twice during the midway point in the first half as Eden Hazard headed Oscar's cross over the crossbar before Pedro's effort took a deflection over Hugo Lloris's goal moments later.
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Before the interval, Spurs pushed for an opener, but Son's speculative shot from distance was well off-target.
After a cagey start to the second period, the former Bayer Leverkusen attacker was in the heart of the action again as he received a lobbed pass from substitute Erik Lamela, only to see Begovic keep out his half-volley.
With a quarter of the game to go, Chelsea had their best chance of the game after Branislav Ivanovic's crossfield pass caught out Kyle Walker, who did not see Hazard creep around the back and volley towards goal, but the Belgian's strike was scooped away by Lloris.
Spurs defender Toby Alderweireld was almost caught out in the dying minutes, but Pedro's chance was quickly snuffed out by the Lilywhites backline before the Spaniard could get a shot off.
However, after Willian wasted a late chance to test Tottenham from a free kick, both teams were forced to settle for a share of the spoils, which means that Arsenal will finish the day in fourth, regardless of the outcome of their game at Norwich City.