With Manchester City's clash with Sunderland postponed due to the weather and Chelsea failing to beat West Brom a day earlier, victory would have taken Arsenal to the summit.
However, in a game where neither side offered enough offensively to win, Arsene Wenger's men were only able to move within a point of leaders Chelsea and two ahead of City.
David Moyes' side, who were frustrated by a late Fulham equaliser on Sunday, have now won just one of their last five Premier League outings and remain seventh.
Former Arsenal striker Robin van Persie almost put the visitors ahead after just two minutes, dispossessing Mikel Arteta before firing straight at Wojciech Szczesny.
In a lively opening, Jack Wilshere was then well tackled by Patrice Evra, having looked to have fashioned a chance on goal, before Olivier Giroud shrugged off his marker and headed wide from a corner.
Van Persie should have tested Szczesny again after 18 minutes but he sliced an effort way wide of goal after Juan Mata's throughball - much to the delight of the home fans.
Both sides were let down by poor touches in the final third, with Giroud again getting on the end of a corner to head wide after 24 minutes.
With neither side creating many further chances, the only other incident of note before the break was an aerial collision between Rafael da Silva and Giroud.
The Manchester United man landed awkwardly after challenging with the Frenchman for a header, and was replaced by Rio Ferdinand at the break.
While the first half ended at fast pace, the second began slowly - with both defences easily able to cope with what was thrown at them.
The first chance of the half went to Arsenal, with Laurent Koscielny's header cleared off the line by Antonio Valencia from a Santi Cazorla corner a minute after the hour.
Arsenal piled on the pressure from there, with Giroud nudging Bacary Sagna's cross marginally past David de Gea's right-hand post 14 minutes from time before Van Persie's header was tipped onto the crossbar by Szczesny three minutes later.
De Gea was called into action again once more before full-time, getting a strong right hand to Cazorla's effort before tipping it around the post.
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