Both sides passed up the chance to go third in the table after a dismal night's action at Central Park.
The visitors came into the game in buoyant mood after their surprise 5-1 win over Forfar Athletic at the weekend - though Cowden produced a decent show in the cup themselves, losing 3-2 to second division Dumbarton.
Both teams served up a poor first half, with Cowden restricted to route one stuff throughout. Montrose probably had the best chance in the ninth minute, when Kevin Webster's cutback was instinctively met by Craig O'Reilly, though his shot hit keeper Andy Carlin.
Cowden rarely troubled Montrose keeper Jim Butter, with a few Darren Gribben efforts all they had to show for.
Indeed, Martin Wood's back header over the bar after 40 minutes could have given them the visitors the opener.
The second half was along the same lines although Cowden started the better, Iain Mauchlen's low shot hitting the sidenetting seven minutes in.
A Matt Slater header on the hour mark sailed wide for Montrose before, at the other end, Paul Doyle and Butter somehow scrambled away from Gribben.
From then on in, it was a question of both sides cancelling each other out, though Cowden maybe should have had a penalty five minutes from time.
Dene Shields looked to have been tripped by Doyle, but referee Chris Boyle waved away the claim.