The Blades looked down and out after Forest put themselves in firm control of the two-legged semi-final thanks to goals from Jack Colback and the impressive Brennan Johnson.
But Berge headed home in injury time to give United fans hope heading into Tuesday’s second leg.
“It’s half time, there’s still loads to go,” Heckingbottom said.
"You could see the whole body language of everyone connected with Forest more so than us, we’re not going to come off the pitch happy we’ve got beat, but it certainly changed the mood amongst the fans, the mood among both camps.
"Hopefully it proves to be a pivotal moment.”
Heckingbottom spoke of his disappointment with Sheffield United’s first-half display.
He said: “I thought we started well, there were still good moments in the first half but every time we lost the ball they were a threat on the counter which we spoke about, prepared for, and we didn’t deal with it well enough.
"We had to be better second half, which we were, however we couldn’t break them down when they were sitting in. They defended their box very well, we want to be better in and around those areas.”
Heckingbottom added: "One goal then would have changed the tie. It didn’t happen, so we have to take the one we got at the end.
"The game might be exactly the same at the City Ground, although it might make for an edgier atmosphere for them if they are going to defend deep or get pinned in by us.”