The Croatian internationalist scored late on with the aid of a deflection, and was a constant threat down the left with a number of superb deliveries into the area.
Despite that, he was keen that the focus went on the team and told RangersTV: "I am very happy myself, but as I have said a lot of times, it is not about one man and you can't play good if the team is not playing good.
"We are playing good, we are in good shape and in good form - we still have a lot of things to improve but I think that is normal.
"We can't just overnight start to play perfectly but the competition is bigger, the squad is bigger and what we are doing in training, you can see.
"So I am very happy that the team played very well.
"I am very happy and very proud because [with the goal] and that I am helping the team to win, but as I said, the most important thing is the team playing good - every time someone will jump and be man of the match, but the most important thing is we are taking three points."
On the game itself, Barisic continued: "It was a hard game but in the end we deserved to take the three points. We had a lot of chances that we didn't score and we conceded from a set piece.
"Of course, in that moment the game was getting harder and harder but we scored the goal and in the end we deserved these three points."
Gers were particularly fluent towards the end of the first half, where Malik Tillman eventually got the breakthrough.
Prior to that, the Light Blues had countless opportunities to score from some brilliant play, and Barisic added: "It was great - but in that situation it is not great if you don't score. We didn't score a couple of good chances I think we should have scored and then sometimes you can concede a goal - that's what happened to us.
"Of course, then we started to become nervous and everyone became nervous but I think that we kept playing well and stayed positive and came back to score the goal and get the three points."