Meszaros scored both goals on shots from the point, beating David Aebischer with 1:13 left in the first and putting another long drive past the goalie on a power play 5:43 into the second to make it 4-1.
Peter Schaefer, Mike Fisher and Chris Neil also scored for Ottawa (5-3). Martin Gerber made 31 saves.
Alexei Kovalev and Tomas Plekanec scored for Montreal (2-5), which had won two in a row, including a 7-3 win over the Senators on Monday in Truro, Nova Scotia, after opening the preseason with four straight losses.
Schaefer opened the scoring 8:44 in when he beat Aebischer from the edge of the crease with a backhander after stopping a pass with his left skate.
Kovalev tied it at 1 at 11:14 when he flipped Koivu's rebound over Gerber.
Fisher put Ottawa up again with a power-play goal at 13:31.
After Meszaros' two goals, Plekanec drew Montreal within 4-2 when he banked a shot off Gerber's left skate from behind the net 8:06 into the second.
Neil restored the Senators' three-goal lead at 12:36, finishing off a set up by Dany Heatley and Jason Spezza.