The Dutchman's first chance of the match arrived in the 13th minute when Theo Walcott picked him out inside the area. However, Van Persie saw his volleyed effort strike the base of the post.
Arsenal hit the woodwork again in the 42nd minute when centre-back Koscielny headed Van Persie's lofted pass against the crossbar.
Chelsea's best opportunity of the half materialised in the 44th minute when captain John Terry headed wide from six yards out.
Before the half was brought to an end Van Persie had another chance, but he drove his effort straight at Petr Cech.
The Chelsea goalkeeper continued to thwart the Arsenal skipper after the restart, denying him on two further occasions.
Daniel Sturridge looked to have been presented with the chance to win the match for Chelsea five minutes from time, but Koscielny retreated to deny the England international a clear goalscoring opportunity.
The point sees Arsenal remain third in the Premier League table, while Chelsea stay sixth.