Potts' goal – on his 150th league appearance for the club – sentenced Blues to an 11th home game without a win, less than 24 hours after chief executive Xuandong Ren gave Karanka a vote of confidence.
Defeat left Birmingham with one victory in their last 12 Sky Bet Championship outings, with three points out of 36 to lie second from bottom of the table, a point below the safety line.
The visitors edged a forgettable encounter, with defences largely on top and goalmouth action thin on the ground as Blues extended their record as the second tier's worst home team, with nine points from 45 at St. Andrew's.
Luton, who ended a three-match winless spell in the Championship to climb to 14th, created the first chance when James Collins beat George Friend to a long pass from Tom Lockyer and lobbed the ball into the side-netting.
Blues' only chance of the opening half-hour was a poor shot by Jeremie Bela that bounced well wide after Kristian Pedersen's cross was headed back into the danger area by Ivan Sanchez and Lockyer touched the ball into the winger's path.
Luton had been the better side and took a deserved lead in the 31st minute.
Birmingham failed to deal with a long throw-in from Harry Cornick on the left, Etheridge made a superb instinctive save to block Collins' close-range shot on the turn and Potts headed home the loose ball.
It was Potts' first goal since October 23, 2019 and only Luton's seventh away goal of the season.
Luton remained on top and Jordan Clark forced Etheridge to tip away a left-footed drive before Potts headed wide from Dewsbury-Hall's corner.
The home side made a triple substitution in the 69th minute in an effort to drag themselves back into the game.
But it was Luton who created the next chance as defender Sonny Bradley sent an overhead-kick wide after Blues struggled to deal with another long throw-in from Cornick.