Mark Byrne and Luke Gambin were on target for the Bees, who stayed two points clear of the relegation spots with victory against an out-of-sorts Fleetwood side who have only won once in their last eight games.
Town's dismal defeat at Underhill came after Saturday's 2-1 reverse at Plymouth, another side battling for League Two safety, and leaves the club in ninth place, six points off the play-offs.
Alexander said: "That was not good enough, it was an embarrassing performance and we were well below the level you should expect from professional footballers.
"You can have all the skill and talent in the world but it's second in the end to your mentality. You have got to prepare for every game in the right way.
"We did not turn up in that game. We have shown we can beat the best teams in the league but it's not good enough when you can't beat those near the bottom.
"The players have let themselves and the people of Fleetwood town down. It's not me they're letting down but the club and what they are wearing on their shirt.
"We prepare for every game the same way. It's all down to hard work now and we will come back on Thursday and keep working hard to see where we end up."