United are now on a nine-match winless run in all competitions — their longest streak without a victory since 1988 — after second-half goals by Morgan Rogers and Cole Palmer on Saturday.
It turned out to be a bad day for Manchester all-round as leaders City also fell at home to Brighton 1-0 for a second-straight league loss after opening the campaign with seven wins in a row.
Carrick was already under huge pressure — and seemingly operating on a match-by-match basis — heading into the game at Stamford Bridge, for which he restored Bruno Fernandes to the starting line-up after dropping the captain last week.
"Things have gone against us, and gone against us in a big way," Carrick said. "It unravelled quickly."
Palmer converted a 64th-minute penalty awarded after Martinez jumped with a raised elbow toward Wesley Fofana at a corner and struck the Chelsea defender in the face.
The Argentina defender, already in the spotlight for criticising United's hierarchy recently, was given a second yellow card in a four-minute span.
As Martinez walked off the field and down the tunnel, Carrick didn't even acknowledge the player.
"It has a major importance on the game," Carrick said of the red card. "A huge moment."
United stayed on just one win in the league after nine games and are one place above the relegation zone.
United's only two victories from 13 matches in all competitions this season were against promoted Ipswich in the Premier League and tournament newcomers Sabah in the Champions League.
Manchester City striker Erling Haaland had scored in his previous seven league games — leaving him four off Leicester great Jamie Vardy's record of netting in consecutive matches — but that run ended against Brighton in a rematch after their League Cup meeting on Wednesday, won by City on penalties.
Yasin Ayari, one of the Brighton players to miss the team's shootout loss, made amends with an 11th-minute winner at Etihad Stadium.
With City also losing at Aston Villa 2-1 last weekend, Arsenal, tied on points in second place, can move top at Liverpool on Sunday.
Brighton, the league's surprise package this season, moved into third place — two points off the lead.
By beating Leeds 2-0 with goals by Evanilson and Amine Adli, Bournemouth made it six-straight victories in all competitions without conceding a single goal and moved up to seventh.
Sunderland stayed in last place after losing 3-1 at Coventry, who leapt out of the relegation zone, while next-to-last Nottingham Forest lost at Brentford 2-0 and promoted teams Hull and Ipswich drew 2-2, with Hull's Australian youngster Lucas Herrington dropped to the bench.
