An unmarked Abraw netted the only goal in the 32nd minute with the simplest of headers after Siphiwe Tshabalala caught the Aces defence napping and played a short free-kick to George Lebese, who turned and crossed for the striker to finish.
Amakhosi coach Steve Komphela named an unchanged team from the one that beat AmaZayoni 2-1 in the Telkom Knockout quarter-finals at the weekend, the same team he said failed to kill that game off.
And they did the same thing in this game as they pushed for a second goal immediately after going ahead but were unable to find the final passes.
Aces grew in confidence the longer the game went on and gave it everything they had in the last 20 minutes, and Chiefs had Itumleng Khune to thank for a string of saves that maintained their slender advantage.
Khune had to make three saves in the space of three minutes between the 75th and 77th minute to keep out a deflected cross right on the goal-line, a close-range header from Collins Mbesuma and then a long-range strike from Lehlohonolo Nonyane.
All three times Khune was on top form, and he singlehandedly saved three points in the fourth minute of stoppage time when he stayed big to save Judas Mosemaedi's shot from a tight angle on the right.
The result moves Chiefs up to fourth place, a point behind Aces and two points behind Absa Premiership leaders Bidvest Wits.