The Gunners slip up allowed north London rivals Tottenham to remain top of the table on goal difference after they won 2-0 at Bournemouth.
United were looking to kickstart their season after losing 2-0 to Tottenham last weekend, but could not have got off to a worse start at Old Trafford.
Taiwo Awoniyi burst clear from a United corner to score for the seventh consecutive Premier League game after 90 seconds.
There were just three minutes and 45 seconds on the clock when Willy Boly then found the net from Morgan Gibbs-White's free-kick.
United escaped with a 1-0 win over Wolves in their opening game of the season despite being outplayed by the visitors to dampen expectations they could challenge for a first league title in a decade.
But on Saturday they battled back to secure a vital three points in thrilling fashion.
Cristian Eriksen started the comeback when he turned home Marcus Rashford's cross.
Casemiro missed an great chance to level before the break but made amends when he fired home Bruno Fernandes' knockdown.
Forest captain Joe Worrall was then sent off 23 minutes from time for bringing down Fernandes as he bore down on goal.
The new United captain was criticised for his body language in defeat at Spurs, but Fernandes won the match from the penalty spot after Rashford was brought down in the area.
"It wasn’t the perfect start for us but we kept control, kept the ball and started doing the right things," said Fernandes.
"The team showed great spirit, passion and desire. We just need to be aware that we can’t concede so early."
Arsenal blow lead
Arsenal made a similarly disastrous start when Andreas Pereira pounced on Bukayo Saka's poor pass to catch Aaron Ramsdale off his line in the first minute at the Emirates.
Fulham's rearguard held out for more than 70 minutes, but their resistance was finally broken when Saka levelled from the penalty spot.
Two goals in two minutes looked to have turned the game around as substitute Eddie Nketiah slotted home Fabio Vieira's inviting cross.
Fulham's Calvin Bassey was then sent-off for a second bookable offence.
But the 10 men salvaged a point when Joao Palhinha swept home a corner three minutes from time.
Tottenham on top
Earlier, Tottenham maintained their bright start under Ange Postecoglou as James Maddison and Dejan Kulusevski struck to see off Bournemouth.
Postecoglou's men have so far shrugged off the departure of the club's all-time record goalscorer Harry Kane to Bayern Munich with new signing Maddison at the heart of most of their good work.
The England international's first Tottenham goal on 17 minutes was just reward for the visitor's dominance of the first half.
"There wasn't anyone happier than me when we got him," said Postecoglou.
"I was delighted. I wouldn't say I'm surprised, but I'm overjoyed at the footballer I've got."
Bournemouth looked the more likely to score in the second period before Kulusevski steered Destiny Udogie's cross into the far corner just after the hour mark.
At the other end of the table, Everton's miserable start to the season continued as Sasa Kalajdzic struck a late winner to secure Wolves' first points of the campaign with a 1-0 victory at Goodison Park.
Brentford remain unbeaten but were denied by Joachim Andersen's equaliser for Crystal Palace in a 1-1 draw.
