Arbroath were first to threaten after 10 minutes when Alan Brazil pulled the ball back to Willie Martin and Chris Tawse made a fine saving tackle to clear the danger.
Arbroath opened the scoring on 18 minutes when a Jay Stein free-kick was headed down by Robbie Raeside and Martin headed home from close range.
On 33 minutes Scott Michie pulled the ball back to Davidson and his drive just cleared the crossbar.
The game boiled over on the stroke of half-time when there was a mass flare-up involving nearly every player on the park.
Arbroath goal-scorer Willie Martin found himself sent off for his part in the melee and less than minute later Montrose were also reduced to 10 men when Rodgers was red-carded for a challenge on Stein.
Early in the second-half Arbroath hit on the break and Jamie Bishop shot just wide of the target.
On 50 minutes a shot from Brian Scott struck the Montrose post.
Arbroath continued to look the more likely side to score and Roddy Black had a 25-yard drive saved then fired another shot just wide of the post.
On 80 minutes Ian Dobbins had a 30-yard drive saved by Andy Reid then at the other end Stuart Cumming shot over from 18 yards.
The final chance fell to Arbroath on 84 minutes when Brazil fired an angled drive just wide of the target.