The deadlock was broken five minutes in when Bobby Barr converted from the penalty spot.
The visitors secured their passage into the next round with two minutes remaining on the clock.
Patrick Walker made a run at the Lossie defence before making no mistake with a close-range finish.
Lossiemouth: Masterton, Sim , Flett, Kevin Stuart, Milton, McMullen, Anthony Ross (Macdonald 57), Urquhart, Scott, Nikki Smith, Main. Subs Not Used: Donaldson, Sean McIntosh, Dan Smith, McKendrick.
Booked: Nikki Smith.
Albion: Gaston, Reid, McGowan, Canning (McKeown 25), Benton, O'Byrne, Boyle, Tyrrell, Barr, Walker, Pollock. Subs Not Used: Ferry, Stewart, Ewings, O'Boyle.
Booked: McGowan.
Goals: Barr 5, Walker 88.
Att: 235
Source: BBC Sport
Source: BBC Sport