A new era kicked off for the League One outfit after a five-year wait to return to Meadowbank ended with the visit of Dick Campbell's side on Premier Sports Cup Group E duty.
Callum Crane wrote his name into club folklore when he fired the hosts ahead in the 24th minute in front of a capacity 499 crowd before Nicky Low restored parity from the penalty spot six minutes before the interval.
A superb brace from Bobby Linn won it for the Red Lichties after the break to all but seal their place in the knockout stages of the competition.
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The Citizens, meanwhile, are fourth in the group with Wednesday's visit of Lowland League newcomers Cowdenbeath before a trip to top-flight St Mirren next weekend still to come.
"We're disappointed with the second half and they maybe just wore us down in the end," Maybury explained. "We're not going to be judged against the likes of Arbroath who finished second in the Championship last season. They're still part-time, they work hard and they're physically a bit stronger than us. They showed more composure than us and they're very good at what they do.
"The goals are preventable but also when we did get the chance to break, we didn't show enough composure. There are things we need to improve on but we're still a couple of weeks from getting there. I know where we're at but we do need to be a little bit better than we were."
He continued: "It's trying to get the balance and right amount of minutes into the players. I knew what team I was going to play today before the Airdrie game during the week and I know what I'm going to go with for Wednesday's game against Cowdenbeath. We want to do well in the League Cup but I've got to make sure that come the first league game of the season at Kelty everyone's had minutes."