United claimed all three points at Stamford Bridge thanks to Matheus Cunha's clinical first-half finish - their only shot on target - in a disciplined display that underlined their growing momentum at a decisive stage of the season.
Carrick was particularly pleased with how his players coped with disruption at the back, with injuries and suspensions forcing a makeshift central defensive pairing. Late changes, including Leny Yoro's withdrawal, added to the challenge in the build-up to kick-off.
"It was a big, important win for us," Carrick said. "We had to deal with suspensions, injuries, and a late change, so the way the players adapted deserves a lot of credit.
"The coaches also did a great job preparing the team over the last couple of days and making sure everyone was ready. There's a lot to be proud of."
Despite limited attacking opportunities, United remained composed and efficient, managing the game well after taking the lead and frustrating a Chelsea side that struggled to create clear chances.
The victory cements United's position in the top five with just five games remaining, putting them firmly on course for a return to Europe's elite competition.
Carrick's men have now built a platform at a crucial point in the campaign, showing both tactical discipline and mental resilience - qualities their manager believes will be vital in the run-in.
With confidence growing and results going their way, United appear well placed to finish the job, while Carrick's calm leadership continues to guide them through a demanding stretch of the season.