The game had originally been due to go ahead at the Marr College playing fields due to waterlogged pitches at the club, but the first team pitch was pronounced playable half an hour before kick-off.
The home side made a lively start, forcing a series of penalties in the Marress half which the visitors did well to survive.
The visitors responded with their first attack of the match, and John Murdoch almost crossed in the corner only to be tackled short and concede a penalty.
Marr took control of the game from this point on, but the visitors took the lead against the run of play on 27 minutes when Cooney scored the first of his double after taking a pass from Martin Sneddon.
The hosts finally scored the try their first-half possession merited four minutes before the interval, but Marress bounced back in front within minutes of the restart. A good driving maul by the forwards forced a penalty, from which prop Ewan Callaghan forced his way over from close range.
The game became increasingly bogged down in midfield in the ensuing minutes but the visitors were given a scare just before the hour when captain Craig Maxwell needed all the gas in his tank to beat his opponent to the ball on a dangerous kick play.
Marress stepped up the impetus in response to this letoff, and on 66 minutes player-coach Scott Hunter drove over on the right after a good spell of possession in the Marr 22.
The visitors' grip on the game was almost total by this stage, and after Maxwell had twice nearly scored on kick plays only for the ball to run dead, Cooney had the last word with his second of the afternoon in the right corner.
Marress are next due to entertain Renfrew before resuming their league campaign at Paisley on January 12.