Two late goals eased Morton's anxiety after Albion made significant changes at half-time and put the home side under much pressure during which Derek Collins and his fellow defenders had to be at their very best.
Morton should have been four up at half-time. Slick passing between Chris Millar, Paul Walker and Peter Weatherson set up a chance for Walker and he strode into the box before slotting the ball under Myles Hogarth from 12 yards in ther ninth minute.
Stewart Greacen nearly doubled the lead with a header from a John Adam free-kick but Hogarth deflected the ball and Alex Williams couldn't convert from near the line.
Williams shouted for a penalty when Stewart Devine challenged him in the box as the striker fed on a splendid Weatherson pass, but referee Cassidy turned a deaf ear.
In the best move of the half, Walker released Williams down the right. Williams held the ball up until Walker had hared up in support. Walker then took over, firing in a savage angular shot which Hogarth saved smartly at his near post.
Albion improved after the half-time changes, but David McGurn in the home goal had only an awkward lob to tip over as Collins, Greacen, Stuart McCluskey and Dean Keenan defended with resolution.
Eight minutes from time Jason Walker came on and, a minute later, he ran on to an Adam pass and cleverly poked the ball beyond the advancing Hogarth.
A minute later, Weatherson sent Millar scampering the whole length of Albion's half to round the exposed Hogarth before tucking in number three.
HASH(0x8c570c4)