In a do-or-die playoff game at Salt Lake Stadium, ATKMB were in cruise mode in the first half and in no time they had a 2-0 cushion, with Williams scoring a brace (6th, 30th).
The Aussie completed his hat-trick with five minutes remaining, but it came after much drama, even as the home crowd of around 32,000 breathed a sigh of relief.
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Six-time Premier League champions Bangladesh, who looked lackluster in the first half, showed they were no pushover as they found a way back into the game with less than 30 minutes left to play.
Costa Rican rookie Daniel Colindres slipped the ball easily past an out of place Amrinder Singh to reduce the margin.
Suddenly the home favorites started to buckle under the pressure as Abahani made promising raids. Their substitute Jewel Rana nearly scored from the right side of the box but his shot went just inches wide.
Moments later, Nurul Faisal curled a cross from the left flank towards Rakib Hossain inside the box but his shot went wide.
The Mario Lemos-coached side continued to threaten, although ATKMB’s Colaco and substitute Kiyan Nassiri missed two easy chances.
Nothing stood in their way, it was Frenchman Hugo Boumous who produced a magic ball from midfield, and the Australian swooped into an open net to seal the issue.
ATKMB thus joined Gokulam Kerala, Bashundhara Kings (Bangladesh) and Maziya Sports and Recreation (Maldives) in Group D of the AFC Cup main round.
They will face Gokulam Kerala in their group opener on May 18.
Earlier, the home giants were given a big boost with an in-form Amrinder Singh, who struggled, back in goal, while Liston Colaco also returned to attack.
With Colaco back on the left wing, the urgency was evident early on as he launched a all-important move to set him up for an advancing Kauko.
The Finnish midfielder drilled a clinical cross towards Williams inside the box, who slammed the Mariners header.
Colaco gained confidence after the early onslaught as he made brilliant raids from the left flank, giving the Abahani defense a hard time.
Prabir Das controlled play from the right flank and almost gave Kauko a blast into an open net, but his shot went over the crossbar almost halfway.
It was Das who set up Williams with a long ball and the latter brought his brace as ATKMB looked unstoppable.
The home side were leading 2-0 at the break but could have scored at least two more by then.