Stephen Humphrys pulled one back deep in stoppage time but City were made to pay for missed opportunities from new recruits Kayden Jackson and Ethan Wheatley as they went down 2-1 at Kenilworth Road.
“They took their opportunities and we didn’t take ours,” said Alexander.
“We started both halves really well but the first goal came from us being disconnected and not committing to being with our teammate. That’s where they find the space but it’s still a great finish from distance.
“We were so on top in the first period of the second half and then we turn the ball over in the middle and they score on the counter.
“We had other opportunities after that to get an earlier goal back that could have built more pressure on them when there’s a one-goal advantage.
“The players keep going to the end but the goal was possibly too late, although we still had a couple of set-pieces after that. We just couldn’t find that killer instinct.
“It’s a game we’ll look back on and rue opportunities for at least a point that we deserved from the overall game. But Luton showed the quality when needed.”
Jackson had a gilt-edged opening with a volley straight after the restart but failed to make contact from close range.
Alexander added: “You’ve got to strike in those moments or maintain (the pressure). What you can’t do in those moments is concede the second goal.
“The first puts us under pressure in our minds to be a little bit open for the second. We’ve still got to have our focus on if the ball gets turned over the opposite way.”
Wheatley should have scored with virtually his first touch when he came on after being played clear by fellow sub Antoni Sarcevic.
“You want to come on and have a positive start. If you’re clean through, one v one – I just think the shot was a little bit like a couple of actions defensively where we didn’t fully commit to it.
“I watched him finishing yesterday in training and he’s putting his foot through it, left and right. We have to find that quality we have from Monday to Friday and put it into action in the pressure situations.”