Albion came close to doubling their lead soon after when Mark Canning's free-kick just cleared the crossbar.
McKenzie saved from David McFarlane after the break and Bobby Barr fired over for the home side.
Albion: Gaston, Reid, McGowan, Canning (McKeown 46), Benton, O'Byrne, Boyle, Tyrrell, McFarlane (Pollock 84), Walker (McCusker 84), Barr. Subs Not Used: Stewart, Ewings. Booked: Tyrrell, McFarlane. Goals: Moyes 27 og.
Livingston: McKenzie, Brown, Talbot, Husband, Moyes, Watson, Fox, Hamill (Keaghan Jacobs 62), Robert Winters, De Vita, Halliday (David Winters 67). Subs Not Used: MacDonald, Sinclair, McDowall. Booked: Brown.
Att: 485 Ref: C Brown
Source: BBC Sport
Source: BBC Sport