Arsenal struggled to break down Forest and failed to register a shot on target in the first half. Arteta introduced Madueke at half-time, and the winger delivered the decisive moment with a powerful, deflected strike in the 80th minute.
"We had to earn it until the last second. They are a well-coached team and hard to break down. We had to take risks; they are a dangerous team," Arteta said.
The Arsenal manager was pleased with how his players remained patient against stubborn opposition, adding: "We had to manage the emotional part really well. We stayed focused, and the finishers made a huge impact."
Madueke was among several substitutes who helped Arsenal find a way through, with Mikel Merino, Viktor Gyokeres and Jurrien Timber also making important contributions.
Arteta believes his squad depth will be vital as Arsenal balance their Premier League title challenge with European commitments, including Wednesday's Champions League trip to Bayern Munich.
"Some of them played three days ago, and some of them will start the next game, and they know that everyone has to be at it," he said.
"If you are needed after 60 minutes, 45 minutes and sometimes two minutes, we have a squad now that can deliver in moments and we have to use that."
Arteta singled out Madueke for his match-winning contribution but stressed that the collective impact of his substitutes was equally important.
"Noni, Jurrien, Mikel was brilliant when he came on, and Viktor really changed the game; all of them did, and that's what we need," he said. "Noni obviously the highlight because he scored the goal and that's the difference."
Arsenal have now won three consecutive matches since their 3-0 defeat against Brighton on September 19. Although their performance against Forest lacked attacking quality, Arteta was pleased his team found a way to secure the points.
"Sometimes you can get frustrated with teams who are difficult to break down. First half we didn't have a shot on target and that's something we discussed," he said.
"We had another two big ones in the second half, but the only one on target we had was a little deflection and it's in."
Nigerian international Alex Iwobi's versatility and quality have earned high praise from Coventry City manager Frank Lampard, who believes the Fulham midfielder was already a good player before their time together at Everton.
Bradley Barcola is available for Liverpool's Premier League meeting with Brighton at Anfield on Sunday, but Dominik Szoboszlai has been ruled out as the Reds continue a demanding run of fixtures.
Arsenal manager Mikel Arteta could hand Max Dowman an increased role in the first team after Bukayo Saka was ruled out for up to eight weeks with a quad injury.
Manchester United head coach Michael Carrick will be looking for an immediate response when his side host Bournemouth at Old Trafford, after their humiliating Champions League defeat to Como.
