The Gunners, fielding a much-changed side from the one that drew with Chelsea in midweek and with manager Arsene Wenger watching from the stands because of a touchline ban, trailed 2-1 at halftime after Eric Lichaj scored twice.
Penalties by Ben Brereton and Kieran Dowell sealed victory for Forest who sacked manager Mark Warburton a week ago.
Per Mertesacker had equalised for Arsenal in the first half and Danny Welbeck pulled Arsenal back to 3-2 down with 12 minutes left but the Premier League side could have few complaints at a bitter exit.
"We played a very good Forest team who were sharp and focussed. The challenges were decisive. Overall our performance wasn't good enough," Wenger, whose side play Chelsea in a League Cup semi-final first leg next week, said.
A youthful Forest, with seven players aged 23 or under, ended the tie with 10 men after Joe Worrall was shown a red card near the end for a foul on Chuba Akpom.
It did little to detract from the party atmosphere alongside the River Trent as Forest's success-starved fans celebrated a rare day of glory for the former English and European champions.
"This football club should be playing against the Arsenals of this world, and they should be beating them," caretaker manager Gary Brazil said.
There were boos from some of the Arsenal fans at the end after they watched their side concede four goals to lower-league opposition in the FA Cup for the first time since 1908.
It was also Wenger's first taste of a third-round exit.
His decision to play a weakened line-up -- only Rob Holding and Ainsley Maitland-Niles kept their places from the starting side in the impressive 2-2 draw with Chelsea in midweek -- quickly began to look like a poor move.
Lichaj headed in from close range after 20 minutes but the lead lasted only three minutes as Mertesacker finished well when the ball fell to him in the box.
American Lichaj produced a moment of magic to put Forest back in front just before the interval, chesting down before looping a volley over keeper David Ospina.
Teenager Brereton sent Ospina the wrong way from the penalty spot after Rob Holding had fouled Matty Cash in the 64th minute.
Arsenal, missing a host of match-winners including Alexis Sanchez, Mesut Ozil and Aaron Ramsey, were thrown a lifeline with 12 minutes to go when England striker Welbeck punished a mistake by Forest keeper Jordan Smith.
Former Arsenal player Armand Traore was then brought down by Mathieu Debuchy as he surged into the box in the 84th minute and Forest were awarded another penalty.
Dowell converted but Arsenal were furious as it appeared as though he had kicked the ball twice after slipping when he was about to strike the spot kick.
(Reporting by Martyn Herman, editing by Pritha Sarkar and Clare Fallon)
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