The Wasps booked a spot in the next stage of the competition with one game to spare after Saturday’s result.
The hosts took the lead through Quinn Mitchell beating Tochukwu Ogayi one-on-one.
Alloa turned things around early in the second half through two quickfire goals from Luke Rankin and Conor Sammon.
Andy Graham said: “That was the message today was to get us through to the next round and we’ve done that.”
He made give changes from the midweek win at Kelty Hearts – firstly switching to a 3-5-2.
The gaffer made up the centre of defence and he was flanked by David McKay and youngster Leo Easdon.
Kalvin Orsi and Steven Buchanan featured as wing-backs on either end of Kurtis Roberta and Cameron O’Donnell in midfield.
Luke Donnelly knitted the centre of the team with Rankin and Sammon up top.
Orsi had the first effort after McKay’s free-kick went all the way to the attacker; he skied the opportunity.
Rankin did well to control the ball and play in Sammon, but the Irishman couldn’t get a clean connection in it.
Ogayi pulled off an impressive acrobatic save to deny Edinburgh; they appealed for a penalty shortly after but were ignored.
Robbie Mahon proved himself a very dangerous outlet for the hosts, giving young Easdon a good test – Ogayi had another fantastic save to stop him.
A long ball through reached Mitchell and the Wasps keeper came off his line but was beaten in the one-on-one for the opener.
McKay stood firm and pulled off a goal-line clearance to stop Edinburgh from getting a quick second.
Graham continued: “Really disappointed in the first half and the way that we played, it was more to do with the basics we game.
“I felt the aggression, the will to go and get on the ball, the bravery to get on the ball – we talk about it all of the time; bravery’s not smashing into tackles.
“Bravery’s taking the ball under pressure and I don’t feel we did that in the first half at all.
“I was really disappointed, I include myself in that, but the whole team was so disjointed and so unlike us.
“Edinburgh should have been out of sight, it was a brilliant goal-line clearance from Dave McKay to keep us in the game.”
The Wasps came out of the break with much more attacking impetus and put Edinburgh on the back-foot.
Sammon’s strike was saved by Weir and O’Donnell’s corner curled dangerously towards the back post but could not be met by anyone.
Alloa had their just rewards after a brief stramash in the box fell for Rankin – he struck it on the volley to level the scoring.
Orsi, who had teed up Rankin for the equaliser, sent a ball in from deep that met Sammon; his header put the Wasps in front.
From there, the only complaint Alloa could have had is that they didn’t score more – Orsi had a fantastic opportunity to put the game to bed but squandered his one-on-one with Edinburgh’s Mark Weir.
Graham said: “One thing we spoke about at half-time is that if teams don’t put us away when they get the opportunity, I always know we’ll get a response from the group of players and that’s exactly what you got second half.
“It was probably a flip, the wind played a massive part in that, of course it did but at the same time, we looked a real threat second half.
“Our quality was miles better and the balls we were playing up to our strikers, we started getting them to link up a bit more.
“We go and score the goals, we should score more and that’s the frustration from my perspective.
“You’d be fuming at the end if you go and lose one but a proverbial game of two halves but the most important thing was to get through to the next round and we’ve done that.”
It wasn’t to be for Alloa in terms of more goals and for Edinburgh, they huffed and puffed but couldn’t find a way back into the game.
They suffered their first loss in the competition, while Alloa won their third on the bounce.
Graham added: “That’s not a defensive unit, that’s a full team thing – if you’re not winning the first ball when it goes up, you’re going to be defending a lot more.
“Our decision making defensively, our quality defensively was really poor, second half was miles better.
“That’s more like what I want us to look like as a group, as a team.
“It’s difficult as well, we’re playing 3-5-2, we’re not used to playing that as well so it does take boys a wee bit of time to adjust.
“Second half was much more like us and I was really pleased with that performance and that response.
“Edinburgh are a good side, they go 1-0 up, they’ve got their tails up, got really good quality in their team.
“You always get a response out of this group and they gave us that today.”