Both sides missed late chances to secure victory, while the hosts had a strong appeal for a stoppage-time penalty turned away.
While the Gunners remain top of the table, Chelsea do still have a game in hand over them.
Arsenal have wobbled of late, losing to Birmingham and drawing with Manchester United, but welcomed back Leah Williamson, while Stina Blackstenius made her full debut.
They were the better side in the first half and came closest to a goal in the 20th minute when Vivianne Miedema hit a post with one of several efforts.
Chelsea, who had offered little in attack, did not really threaten until first-half stoppage time, when Pernille Harder and Sam Kerr were both denied by last-ditch defending.
Emma Hayes’ side improved after the break and should have taken the lead in the 55th minute but Kerr failed to make contact with Harder’s cross when faced with an open goal before Fran Kirby saw her effort blocked.
Both teams lost players to injury, with Chelsea’s Erin Cuthbert and Arsenal’s Rafaelle Souza both limping off.
Arsenal played themselves into trouble when Manuela Zinsberger failed to hold on to a shot from substitute Ji So-yun and the keeper and Williamson got into a muddle, but Harder could not take advantage.
As the clock ticked towards 90 minutes, Arsenal nearly found the breakthrough twice in quick succession from set-pieces.
First a free-kick caused mayhem in the Chelsea six-yard box but desperate defending eventually got the ball away, before Millie Bright headed off the line from a corner.
Chelsea wanted a penalty in added time when the ball hit Williamson’s arm but the referee was unmoved, while Kerr had a header saved by Zinsberger in the last of eight minutes of added time, as it finished goalless between the two sides.
Additional reporting by PA Sport.