Abraham swept in the winner after nice play with Willian but it was the introduction of Jorginho off the bench in the first half that changed the game.
"We were so awful for 30 mins, slow lethargic, nervous," Chelsea manager Frank Lampard said.
"The opposite of against Tottenham. We gave Arsenal everything they wanted. We made the change early, felt it had to be done. Second half we dominated.
"After 10 mins or so it was clear that Arsenal had seen us play against Tottenham and made us into a back five instead of a back three. They were on top and there was a lack of balance to our team. The sub gave us another man in midfield and gave us more urgency."
Lampard said Chelsea still needed to sort out their home record, a problem emphasized by their loss to Southampton on Boxing Day.
"Our next home games we'll do everything we can to turn the corner," he said.
"We're still in fourth and want more and the home games have been different so we have to focus on them when they come around."
Arsenal defender David Luiz said the side showed signs that they will soon be back up the table under new manager Mikel Arteta.
"It is not easy for him in the middle of the season, he didn't choose the players, but he will do his best," Luiz said.
"That's why we could see Arsenal dominate in the first half with amazing football and quality. We should (have) scored a second goal and then it would be a different game.
"But congratulations to Chelsea. Frank has come in and they're doing what he wants them to do."