Carrick was given a stay of execution by the United hierarchy after a 3-0 home defeat to Bournemouth last weekend when many expected the axe to fall on the former England midfielder.
United were much improved during the first half at Stamford Bridge but paid for a lack of ruthlessness as the visitors wasted a number of chances to take the lead.
Morgan Rogers strolled through the heart of the United defence to open the scoring early in the second half.
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Lisandro Martinez was then shown two quick yellow cards and conceded a penalty, converted by Cole Palmer, to kill off any hope of a United fightback.
"Yes," Carrick told TNT Sports when asked if he expected to remain in charge for the Roma match.
"We've got to move forward. We've got to take the things that we felt worked for us today and take a bit of belief and the confidence from that and a lot of the good things, especially in the first half."
United's longest winless run for 38 years leaves them just one point above the Premier League relegation zone and at risk of dropping into the bottom three by the end of the weekend.
"From being in such a good place in the first half, the two halves of football couldn't probably be more further apart in some ways," added Carrick.
"I think the way they went about it and played, the quality of football and the way we played in the first half, definitely gives encouragement. We've got to take that."
Two defeats in their first three Champions League games have left United 20th in the 36-team table and at risk of missing out on the knockout stages.
