After a goalless first half, Chelsea broke the deadlock in the 40th minute through Morgan Rogers. Romeo Lavia won possession in midfield and the England international drove towards the United defence before producing a clever finish, lifting the ball over goalkeeper Senne Lammens.
United's hopes of recovering suffered another major blow when Martinez was sent off with 63 minutes played. The Argentine defender received a second yellow card after barging into Wesley Fofana during a Chelsea corner, with VAR confirming a penalty.
Cole Palmer converted from the spot to make it 2-0, despite Lammens getting a hand to his effort. The goal effectively ended United's hopes of rescuing a result, with Chelsea taking control against their 10-man opponents.
Speaking after the clash, Carrick told reporters: "I think the first half was as good as we have played for some time. We looked dangerous, we looked fluid and we looked like a really good team.
"We could and should have scored one, two or even three with the space we created - but then the game gets away from us."
"I’ve not seen it [the red card] back, so I don’t know if it was or not," he added. "The referee has made his decision and it’s a big moment in the game, and then the second half gets away from us. It’s something we have to deal with. Sometimes you get setbacks in games and you have to deal with that. The two halves of football couldn’t be further apart."
With the three points, Chelsea moved up to fourth place on the league standings with 16 points from nine matches. United remain in 16th place with seven points.
"You’ve got to keep moving forward and you do that by winning games, it’s as simple as that. There are no words I can say to you now that will make that any different. You’ve got to move forward and win football matches," Carrick explained ahead of hosting Roma in the Champions League on Tuesday.
"I’m looking forward to getting the boys ready for midweek now.
“The supporters have been absolutely amazing. Not just turning up and the effort that’s needed, but they are spending their money and they have been really tested.
“They’ve had some really big highs over the years, but it’s easy to go the other way. The way they have stuck by and supported the team, it sums up what they’re all about. We’ve got to give them something to really get their teeth into, like we did in the first half," he said.
Manchester United's crisis deepened after they suffered a 2-0 defeat to Chelsea at Stamford Bridge, extending their winless run to nine matches in all competitions and increasing the pressure on manager Michael Carrick.
Youri Tielemans has thrown his support behind Michael Carrick, insisting Manchester United need stability rather than another managerial change as they attempt to turn around their poor start to the season.
Tottenham have received a fitness boost ahead of Saturday's Premier League clash with Crystal Palace, with James Maddison available despite suffering an injury scare against Chelsea.
Chelsea head coach Xabi Alonso has insisted Cole Palmer's future at Stamford Bridge was never in doubt, dismissing suggestions that the England international was close to joining Manchester United.
