Stanley named the same side that won 4-1 at AFC Fylde in the FA Cup on Saturday and Dunbavin had to make a fine save from Jack Redshaw in the first half.
Lewis Alessandra and Jack Redshaw then both hit the bar after the interval as Morecambe attempted to break the deadlock.
Stanley substitute Osayamen Osawe had a shot saved before Dunbavin saved Richard Brodie’s penalty after Dean Winnard had fouled Kevin Ellison.
Stanley: Ian Dunbavin, Dean Winnard, Peter Murphy, Rob Atkinson, Michael Liddle (Osayamen Osawe 66); Luke Joyce, George Miller; Rommy Boco, Will Hatfield (Charlie Barnett 56), Lee Molyneux; Padraig Amond. Subs not used: Andrew Dawber, Toto Nsiala, Craig Lindfield, Bohan Dixon, Jack Sampson.