Manchester United threatened before half-time but failed to convert their chances. Matheus Cunha had an effort ruled out for offside, while Bruno Fernandes and Amad Diallo missed promising opportunities. Bryan Mbeumo later struck the crossbar as United chased a response.
Michael Carrick felt his side deserved to lead at the interval.
"We were more in the game first half. Felt we should have been ahead," he told Match of the Day. "I was happy with the team first half, especially individually in it. The discipline, the role, the quality of the football were all there."
Chelsea improved after the restart and broke the deadlock five minutes into the second half. Roméo Lavia won possession and supplied Rogers, who drove through the United defence before lifting a composed finish over Senne Lammens.
Carrick was unhappy with the goal.
"The first goal is not a good goal to concede. After that, we never got the game to where we got in the first half," he said. "We didn't come out, and didn't get going in the second half."
Alonso said his half-time instructions had helped Chelsea take control.
"We spoke at half-time that we were going to win the game with the ball. The message was to be better with the ball. We changed some things tactically. We had more control."
The contest swung decisively in Chelsea's favour just after the hour. Martínez, already booked, was penalised following a VAR review for a challenge on Wesley Fofana inside the penalty area. He received a second yellow card and was sent off, while Chelsea were awarded a penalty. Palmer converted from the spot in the 64th minute.
Carrick recognised the significance of the dismissal.
"It has a major importance on the game. A huge moment. I think that is pretty obvious really."
He had not reviewed the incident, but said he did not feel let down by Martínez.
Alonso praised Estêvão, who threatened throughout and had a goal ruled out for offside.
"He was a threat, he created many chances that we could score," he said, also describing Lavia's display as "a top, top performance".
Alonso said Estêvão earned his start after responding positively to being left out against Tottenham.
United tried to respond with ten men, and Mbeumo came close when his late strike hit the crossbar.
Carrick admitted: "There is no hiding from the fact that we haven't won enough games."
He urged his players to take encouragement from their first-half display and confirmed he expected to remain in charge for the next match.