The away side took an early lead when Jack Colback scored on the rebound after Wes Foderingham had kept out Philip Zinckernagel's initial effort.
With time almost up, Berge bundled home from a corner to set up a big second leg on Tuesday.
Forest were just two minutes away from leaving Sheffield with a richly deserved two-goal lead but will still start the second leg as favourites to reach Wembley.
After an even opening 10 minutes, Forest went in front when Sam Surridge got on to Joe Worrall's clipped ball into the inside-right channel and sent the ball into the middle for Zinckernagel. He should have scored but fired straight at Foderingham and, with Blades full-back Ben Osborn having slipped over, Colback was able to send the rebound home under no pressure.
That was as good as it got in the first 45 minutes for the home side though, as Forest totally dominated the rest of the half.
Ryan Yates headed wide from a great position after James Garner's cross found him unmarked in the middle, before keeper Foderingham saved well from Surridge and then did brilliantly to get up and keep out Johnson's follow-up header.
Foderingham had previously made separate saves from Surridge and Johnson, while the latter screwed an effort past the post, as the Blades clung on for the interval.
Paul Heckingbottom's men massively improved after the break and it was the visitors who had to weather a storm.
However, despite dominating the possession they could not force Brice Samba into a notable save and were then caught cold when substitute Joe Lolley robbed Egan of possession 30 yards from his own goal and drove forward, before the ball broke nicely for Johnson to sweep in a quality finish.
It could have got even worse for the Blades but Egan made a great block to deny Johnson after an enterprising run from Djed Spence, and Spence then sent a low drive into the side netting.
With the home fans streaming for the exits, their team won a late corner. Morgan Gibbs-White's centre was missed by Samba and Norway international Berge got the crucial touch to put the ball over the line.
Heckingbottom later said: "It's half-time, there's still loads to go but I think you could see the body language of everyone, connected with Forest more, we're not going to come off the pitch happy that we got beat but it certainly changed the mood.
"Hopefully it proves to be a pivotal moment.
"I thought we started well, there was still good moments in the first half but every time we lost the ball they were a threat on the counter.
"We had to be better second half, which we were but we couldn't break them down, they defended the box very well but one goal then would change the tie, it didn't happen so we have to take the one we got at the end."
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