In front of a crowd of around 150, the home side made a bright start with Alex Rudd seizing on a defensive slip but his effort went wide of the target. Ben Mitchell made a smart stop in the home goal with 20 minutes played as the visitors upped the intensity with extended periods of possession.
Lee Smith then went close for Fairford as chances opened up, with Rudd then firing over a free-kick which failed to trouble Connor Johns in the Robins' goal. Jennings then hit an effort straight at the Swindon custodian before the half-time whistle blew.
Cooke, who played against Everton for the Swindon first team last Saturday, came off the bench for the second half and made an immediate impact. The powerful frontman fired the opener on 53 minutes before Seb Martinez doubled the visitors' advantage a minute later with a superb curling effort.
Cooke smashed home his second of the night on the hour after Jake Evans had seen his shot come back off the post. The striker was then denied again by Mitchell twice, before Evans saw another effort saved.
Fairford reduced the arrears in the closing stages with Danny Hale scoring from the spot.