Martin signed for Coventry on loan from Southampton at the start of the week and was outstanding in a resolute Coventry defence.
Carsley said: "Aaron comes from Southampton so it was almost a local derby for him. He had been with us only for one training session, but he coped superbly.
"Overall we were compact, resilient and played with plenty of courage. We had a game-plan and it worked well from start to finish.
"We had to have that mentality and that comes from going away and winning. We have done that 10 times now.
"I'm still gutted at losing to Crewe on Saturday, but we did not panic here. I told the players exactly what would happen at Bournemouth and how they would play against us and so it proved.
"Our two central defenders Aaron Martin and Nathan Camderon were outstanding, but it was a great team effort."
Leon Clarke outstripped defender Steve Cook to reach a long ball out of defence and then converted his 20th goal of the season with a left-footed shot.
And in the 84th minute Lewis Grabban handled a free-kick in the penalty area and Carl Baker easily converted the spot-kick to complete Coventry's win.