New Aldershot boss Andy Scott only arrived on Friday after the club parted company with Dean Holdsworth and things went perfectly to plan for the hosts, with Danny Hylton putting them ahead after 38 minutes before Fleetwood capitulated after the break with both Jamie McGuire and Alan Goodall seeing red.
Goodall's offence was a professional foul in the penalty area and although Craig Reid's spot-kick was saved, Hylton made sure of the points with his second six minutes from time.
Alexander said: "It's not a day to forget because we'll remember how we feel and if you brush these days under the carpet you're lying to yourself, so we're going to remember this day and use it for the rest of the season.
"My team's commitment in the first half was the most disappointing thing about it. It was nowhere near the levels we expect.
"Football's unforgiving, if you think you can play for half a game and be successful then we're kidding ourselves but we know that wasn't good enough."
Regarding his side's two dismissals, Alexander added: "The second red card definitely wasn't a sending-off, the lad's dived over the 'keeper, the defender hasn't touched him so it wasn't a sending-off.
"The first one was right in front of me. He's raised his hands, he hasn't hurt the lad, you'd think he'd been hit by Tyson, but you can't raise your hands so he can't defend it."