Cove Rangers manager John Sheran conceded the tie was won and lost in the first leg at Harlaw Park in Inverurie when City came away with a 3-0 triumph.
Cove earned a 1-1 draw in the return at Meadowbank on Saturday but Sheran had no complaints and said: “We just can’t afford to give any team a three-goal start, particularly one which plays five at the back, sits in and tries to hit on the break.
“We made ourselves second favourites to progress after that first leg and today we just couldn’t get going throughout this match.
“If Connor Scully’s goal had stood it may have been a different story, and I couldn’t see why it wasn’t allowed, but so be it.
“Edinburgh City’s style of football is not my brand, I think it is negative football, but it works really well for them and they are very well coached.
“They deserve to go through.”
Sheran and his players will try to finish the season in style in Saturday’s Aberdeenshire Shield final against Fraserburgh at Harlaw Park and the manager added: “We have the chance to beat the only team we haven’t defeated this season.
“The Shield is an excellent tournament to win and we will be doing our best to make it a fitting end to our season.”
Edinburgh City manager Gary Jardine was thrilled to have led his side into the final two-legged play-off against East Stirling after losing to Brora Rangers in the semi-final last season.
Jardine said: “Last week’s 3-0 win was a fantastic performance and this was more a case of simply getting the job done.
“We weren’t going to go for all-out attack but it would have been easier for me to prepare this week had it been 1-0 or even 1-1 going into the second leg.
“It was difficult for me and probably more so for the players, should they stick or twist, but I think we got it just about right.
“I don’t think there was any complacency at all.”
Looking forward to the two-legged play-off against East Stirling, Jardine insisted his side will start as underdogs and added: “They have played at a much higher level all season, but we will go and enjoy it.
“It is a sin no club has ever given our captain, Dougie Gair, the opportunity to play his football at a higher level.
“The same goes for the rest of my squad, and they are determined to take this chance against East Stirling.”
