The home side created the better chances in the first half and had the ball in the net just before the break when Mugabi converted following Mark O'Hara's knockdown.
However, Dallas penalised O'Hara for a push as he rose to head Liam Polworth's free-kick.
Robinson said: "The disallowed goal changes the whole game. Hopefully I will get an apology from the referee.
"He said it was a push in the box. I'm not one that comes out and criticises referees, we all make mistakes, but that was a goal. There's no push.
"I'm not saying that's the reason we lost but it certainly changes the momentum of the game.
"Going in at half-time 1-0, Hibs are a good side, they have some very, very good forward players, and they would have to come out and chase the game and that leaves us to expose them at the back. They didn't have to do that."
Robinson accepted the game was ultimately lost in each box.
He added: "We didn't defend well enough in one-v-one situations against their good players like Martin Boyle and they have the luxury of bringing Doidge off the bench, who they have paid a lot of money for. The front boys were probably the difference in the game.
"We had chances, Tony had two great chances, Bevis hit the post and we had a goal disallowed. We got punished for errors and their quality up top told and we didn't quite have the rub of the green."
Hibs head coach Jack Ross revealed a half-time pep talk had produced the desired effect.
He said: "At half-time I was critical of the players in some aspects. We gave ourselves a foundation to win the game but we weren't quite at the levels we need to be to win matches.
"Second half we were terrific in all aspects of our play. So I am really delighted with how the players responded.
"They are such a good group and a humble group that sometimes they need a reminder that they are good players and they should be confident but we also need a degree of intensity about our play.
"We do a lot of work in getting the ball out of tight areas into wider areas and we didn't do that quickly enough in the first half.
"Second half we did it a lot quicker right from the off, hence the reason why we create opportunities throughout the course of the second 45. Just all round our game was so much better second half.
"But although I was critical first half, the one thing we did was hang in and give ourselves a platform to win it."