Everton and Watford both progressed against Championship opponents, the Toffees enjoying a 3-1 victory over Rotherham United and the Hornets winning 2-0 at Reading.
It took Forest two minutes to get ahead against Rafa Benetiz's Newcastle with Daryl Murphy heading in Sam Byram's cross.
Aitor Karanka's side, who played some attractive and positive football, looked set to celebrate a well-earned victory against top-flight opponents but Newcastle substitute Salomon Rondon levelled the game in the second minute of stoppage with a low drive that flew in off the inside of the post.
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Forest responded straight away, though, with a swift break forward that was finished off superbly by a powerful low drive from Matty Cash.
Newcastle were furious after their winger Ayoze Perez went down under challenge from keeper Luke Steele but their penalty appeals were ignored and Forest wrapped up the win with a clever chip from the impressive Gil Dias.
"It was great to win against a Premier League team of their quality and experience," Karanka said.
"We never give in until the end and we showed that again today. We always believe. When we play in our style, we show just how good we are."
Benitez had no doubt that Newcastle should have been given a penalty and felt his team's performance merited a different result.
"Everyone could see what happened," the Spaniard said. "I don't agree that Forest were the better team and I'm not concerned by how we played."
Everton manager Marco Silva made eight changes to his side and one of those introduced, forward Dominic Calvert-Lewin, took full advantage of his opportunity with two goals against Rotherham.
Icelandic playmaker Gylfi Sigurdsson put Everton in front from close range and the two goals from Calvert-Lewin, either side of Will Vaulks's header for Rotherham, ensured progress for Everton.
Calvert-Lewin's second was a beautiful, curling shot from 20 yards out
Watford have made a 100 percent start in the Premier League and despite making wholesale changes they enjoyed a win at Reading thanks to goals from Isaac Success and Domingos Quina.
Championship Millwall beat third-tier (League One) Plymouth Argyle 3-2.
(Reporting by Simon Evans, editing by Ed Osmond)
