Jamie Grimes headed the Spireites in front but two late goals from Dior Angus swung the scoreline around for the unbeaten hosts.
Overall, Cook was disappointed by the Blues’ poor second-half and said the Robins deserved to win.
He told the DT: “I thought we controlled the game in the first-half, but I didn’t think we turned up in the second-half. I was really disappointed with the performance, I was really disappointed with everything about it.
"I thought we would (kick-on in the second-half), I told the players they have to go and score into those fans, they will then give you the energy to keep going, but it never happened like that. Altrincham came on to us a lot more than we thought they would in the second-half and, in my opinion, thoroughly deserved the win.
"All our weaknesses probably came out in the last 10-15 minutes – the softness, the not making good decisions, not retaining possession of the ball, you can write a list and I feel Altrincham fully deserved to win the game.
"We have this Achilles heel and I never feel comfortable. I watch us getting deeper, sloppier, giving balls away and refusing to do the basics like clearing your lines and within that you get punished.
"I thought at times we looked good in the first-half, probably lacked that little bit of penetration that I felt was coming, I thought we would tire Altrincham out, and it actually looked the total opposite. Altrincham finished the game strong and deserved to win.”
He continued: "One of the problems I have is, we had a point, you are deep into stoppage time, yet some of the decisions we made in those final minutes were beggers belief. Just play basic football. You don’t have to do anything great. You just have to turn the opposition around and play in their half. We were giving passes away in the middle of the pitch that results in goals.”
Chesterfield return to action on Monday against Hartlepool United.
Cook added: "I am not going to criticise players, we always keep that in house, but we have a softness about us and that has got to change if we are going to be successful this year.
"We have got a squad of players now that will be utilised.”