But Falcons fly-half Loki Crichton missed a penalty from in front of the posts with the game's final kick, and Worcester could celebrate their third successive league win.
It took them off the bottom for the first time this season, moving one point above Northampton, although the Saints have two games in hand.
A drained Brain said: "I was halfway down the tunnel when Loki Crichton missed that kick. I didn't see it, but I heard the cheer.
"It was a bizarre game in many ways, and we seemed to be in an irretrievable position at half-time, 21-6 down.
"We defended very poorly in the first half against a very good attacking team, but we kept our discipline in the second period and it was a very good fightback."
Worcester have just three league games left - away to Bristol, at London Irish and at home to Saracens, and Brain added: "We've done our bit to put pressure on the teams around us.
"We have won three in a row, although I have to say that was our worst performance of the three. In the first half, we just weren't at the races."
Newcastle, despite being without England trio Toby Flood, Mathew Tait and Jamie Noon plus the injured Jonny Wilkinson, made the most of Worcester's defensive weaknesses as flanker Ben Woods (two) and hooker Matt Thompson scored tries, all of which were converted by Crichton.
Worcester looked down and out at that stage, but Brain's substitutions worked a treat as hooker Aleki Lutui crossed twice before Gavin Quinnell powered over deep into stoppage time.
Full-back Shane Drahm kicked two penalties, but it was Crichton's miss that proved the major talking point.
Newcastle director of rugby John Fletcher said: "That kick would have gone through the sticks 99 times out of a hundred, but on this occasion it didn't.
"But it wasn't just that in terms of the game in general from our perspective. We stopped playing a little bit in the second half and we became indisciplined which hurt us."