Billy Paynter came close for Swindon in the first half, but his shot hit the side netting.
Crewe took the lead just before half time with a close range goal by Shaun Miller on 45 minutes.
Swindon came close to getting an equaliser a few times with Kennedy putting in a number of good crosses, but no one was able to turn them into goals.
Swindon were reduced to ten man on 66 minutes when Billy Paynter was sent off after an off the ball incident.
Overall an average game where Swindon should have at least got a draw, as they had a number of chances, but they couldn't turn them into goals. Swindon's next game is at home to Yeovil next Saturday.