The experienced Town keeper gave referee Jeremy Simpson little choice when he handled Sean McConville's through-ball outside his area with five minutes remaining at Highbury.
Town did have keeper Kieran O'Hara on the bench but couldn't bring him on because all three substitutions had been made.
It meant midfielder Callum Camps putting on the gloves for the closing moment and conceding the winner scored by Michael Nottingham from a corner deep into stoppage-time.
Head coach Crainey said: “Alex’s red card was disappointing of course, and after that I’m just hoping we can dig in and just get a draw from the game.
“It probably is a red card because he’s stopping a goalscoring opportunity.
“But they score from the corner and we’re bitterly disappointed not to get anything out of the game.
“It’s hard to take, especially when you go down to 10 men. I thought we were battling really and competing, which is what I want them to do every single day.”
“But I felt for the group after we conceded in stoppage time like that, especially because I thought they were fighting really hard in that final period to get something out of the game.”
Town will return to Highbury on Good Friday to complete a sequence of three successive home fixtures in League One against Oxford United.