Alexander admitted “some criticism is right”.
"We didn’t compete at the level that we needed to to win the game,” he told the BBC. “We gave away a really poor goal, good quality from the lad. I actually thought we were a bit naive at both ends of the pitch.
“The goal is so early in the game it has put us on the back foot. It is a moment of quality from the lad to stick it in the top corner from there.
"It was the perfect start from them, the worst start from us and allied to the poor goal we gave away in the first leg it makes it a difficult game for us.
"I just thought any momentum we got going at any time was nullified by Sligo in the right moments. We got to good areas, got to good positions but didn't have that killer instinct or that will to beat our opponent in the box.
“We have to take the criticism that is coming our away, we can’t hide away from it. I won't hide away from it, I’m sure the players won't. We have to work hard.
"Maybe a reality check of where we are going into the league season. We’re two games into it and there is a lot of football to be played this season.”
When asked if he felt his players let him down, he added: “No I wouldn’t say that. I back my players all the time. My responsibility as a manager is to lead. If stuff comes my way that’s okay, I understand it.
"I always take responsibility, I’ve been doing it for decades and I’m not going to change my personality.”
Motherwell were backed by 300 travelling fans who voiced their frustration during and after the loss with one supporter’s expletive-laden tirade towards Alexander picked up on the radio.
"We appreciate the support, that’s all I can say,” Alexander noted. “We try our best, it wasn't good enough tonight obviously.”
With three summer signings, the club will look to add before the end of the transfer window.