A moment of quality won the match for promotion chasing Fleetwood as experienced 53rd minute substitute Jon Parkin's defence splitting first time pass enabled fellow striker Jamille Matt to apply an equally good first touch finish.
Alexander said: "One goal was enough, and we kept another clean sheet and every player on the pitch - including those that were on and came off - gave everything for the team."
After a dour, shot-shy first half Matt forced home keeper Martin Rice to save well at the start of the second half.
Recalled Rice also got a touch to deflect Antoni Sarcevic's angled drive from the right off the face of the far post as Town pressed for a crucial opening goal.
Alexander added: "We have come to a club at the bottom of the table but you can see the effort they put in today as well and they made us work hard for the win.
"If we had taken a couple of opportunities early in the second half we could have made it more comfortable for ourselves.
"It was a great bit of skill from Parky to put Jamille in and it was a quality finish. It was Jamille's second opportunity. He had one just previous to that when the keeper has made a good save.
"Sarcevic has hit the post but I think the keeper tipped that one as well.
"We have got that in the ranks. Jamille has unfortunately been injured for four months and we have had to be patient with him but we know he can score goals. We were disappointed with him in the first half and told him at half time and in the second half I think we all saw the real Jamille Matt."