The hosts led through Cauley Woodrow's first-half penalty before goals from Philip Zinckernagel, Brennan Johnson and Lewis Grabban sealed it for Forest.
The opening goal came when Woodrow was fouled in the area by Tobias Figueiredo and Woodrow (20) beat Brice Samba with the resulting penalty.
The equaliser came when Forest sprung the offside trap with Johnson squaring the ball unselfishly to Zinckernagel (61), who tapped into the unguarded net.
The visitors took the lead when Zinckernagel put Johnson (68) through and he slotted the ball past Collins.
Zinckernagel saw his powerful shot saved by Collins before Forest went further ahead.
Alex Mighten did well to beat Jasper Moon on the left and sent over a low cross which was turned in by the unmarked Grabban (82) at the far post.
Cooper said: "It was a game of two halves in terms of the scoreline, but I thought that we actually got into an equal amount of good positions in the first half - we just didn't capitalise on it.
"We didn't deserve to be one-nil down, but we didn't deserve to be one-nil up either, but we could see where the game was there to be won, in behind and down the sides of Barnsley's back line and we were doing that.
"Once you've got in behind you've got to finish it off and I think that was the difference between the first and the second half.
"We continued to get in behind, but we were more productive and scored three goals.
"I haven't seen it (the penalty) back, but it seemed one of those that could go either way.
"The only thing on my mind now is Saturday."
Barnsley head coach Markus Schopp said: "Really disappointing. I think we played well for 45 minutes.
"We came out very intense. We were really active and we were hungry to get a second goal.
"I'm really disappointed because we gave away three points today.
"The two goals were really big punches for the guys and the reaction was not good after that.
"We lost the ball too easy and our second and third line was not enough to counter-press.
"It is individual mistakes, especially for their first goal. When we lose the ball, it must be in a different way and not too easy.
"This is a tough one for all of us. We have to find better solutions.
"Right now, everything goes in a really, really hard direction so we have to be really careful.
"We know it is a tough situation, but we know can do better. Everybody wants more."