Al Atteqi gave the visitors the lead in the semi-final first leg before the break when he converted from the spot after Aws Ibrahim was sent off for bringing down Rogerio in the box.
Kabi then wrapped up victory with a fine individual effort to give the Kuwait side a handy advantage for the home game in two weeks' time.
"Although I was far from the players, they followed the instructions I gave them before the match," said Kuwait boss Dragan Talajic, who was forced to watch the game from the stands.
"Their obedience was the key to the success. I am really impressed and pretty much satisfied with what they showed in the match. They pleased me and the fans too.
"We won outside of our country. I can say that we have passed the halfway mark successfully. I believe we are a lot closer to the final."
Arbil coach Ayoub Odisho, meanwhile, refused to give up hope.
"I understand the next match will be extremely hard for us, but this is how football is. We still have time and the fight is not over yet," he said.
"Despite the difficulties we may face in the coming match, I am still optimistic. The result isn't so bad that we can't come back. We must play better in Kuwait and aim to come back to Arbil with a good result."
The other semi-final saw Nasaf overcome Al Wehdat 1-0 at Karshi Central Stadium.
Jahongir Jiyamuradov notched the winning goal from the penalty spot two minutes before the break after Ivan Boskovic was fouled in the box. The hosts also hit the bar and Boskovic saw a goal ruled out for offside.
"The narrow win gives us an extra but very small advantage but still everything should be decided in the second leg.
"It will be difficult match for us but we are very close to our aim and we have to try our best to reach the AFC Cup final," said Nasaf boss Anatoliy Demianenko.
Al Wehdat coach Mohammed Kwid said: "I believe that we deserved at least a draw today and only one mistake cost us a lot but our chances to qualify for the final are the same that Nasaf have."