Alexander believes Town should have been awarded a penalty in the second half, but he was even more furious with his side's first-half display, during which Billy Kee scored the only goal of the game.
Alexander said: "I thought we should have had a penalty in the second half but there was no excuse for the first 45 minutes.
"We just didn't compete - it was pathetic and it looked as though we didn't want to be here never mind wanting to get through to the next round.
"It wasn't a competitive football side out there, whether it was a Graham Alexander team or anyone else's.
"I can forgive people for misplacing passes or whatever but if you haven't got the desire to compete in a game then you're going to get turned over.
"We tried to turn things round in the second half with more effort and aggression but the damage was already done.
"We thought we should have had a penalty when (Antoni) Sarcevic was taken out - he's adamant about it - but we didn't get it and it still doesn't make up for the way we played in the first 45 minutes."