The Lowland League champions, 3-0 up after the first leg. went further ahead through Dougie Gair's chipped finish.
Cove levelled on the day through Jamie Watt, but the Highland League champions could make up the remainder deficit.
And Edinburgh will place League Two bottom side Shire over two legs for a place in that division..

The writer's work tells the story of a Nigerian woman who moves to the US to pursue a college education.
In 2008, her second novel, Half of a Yellow Sun, won the UK's Orange Prize and Purple Hibiscus was longlisted for the Booker Prize four years earlier.
Other category winners for the US honour included Sheri Fink's book about Hurricane Katrina.
Her account of the patients, staff and families who took shelter in New Orleans' Memorial Hospital during the devastating storm took the non-fiction prize.
Frank Bidart won the poetry section for his collection Metaphysical Dog, while Amy Wilentz was honoured with the autobiography award for her account of journeys to Haiti following the 2010 earthquake in the country.
For the first time, a special award was given for a debut writer, crossing all categories.
Anthony Marra was honoured with the prize for his novel A Constellation of Vital Phenomena.
Adichie's third novel was also named as one of the New York Times' top 10 books of 2013.
In an interview with the Los Angeles Times, the author said her book drew on her own experiences as an African living in the US, particularly with African Americans.
"I don't know race in the way an African American knows raceâ€¦ Sometimes it takes an outsider to see something about your own reality that you don't," she said.
Her preceding work, Half of a Yellow Sun, is set during the Biafran War of the late 1960s and has been adapted into a forthcoming film starring Chiwetel Ejiofor and Thandie Newton.
The writer is also in the running for the UK's Baileys Women's Prize for Fiction - formerly known as the Orange Prize - for Americanah.
The National Critics Book Prize was first awarded in 1974 and is open to writers of all nationalities whose work has been published in the US.

Wayne Rooney and Ashley Young are back in training after respective ankle and groin injuries and might be involved.
Chelsea head coach Antonio Conte has a full-strength squad and could name an unchanged side.
Cesc Fabregas and Willian are the most likely of any players to earn a recall.
Guy Mowbray: "On paper it's one of the biggest fixtures of the season, but there are factors to suggest that this won't be a United-Chelsea classic.
"A close one as with last month's FA Cup tie? Almost certainly. End to end attacking excitement? Probably not.
"Chelsea will be fresher than United and quite frankly are much the better team right now, so Jose Mourinho is unlikely to set his side up for an open game.
"Moreover, he'll be very keen to avoid any prospect of a third defeat of the season to his old side - and maybe more pertinently to Antonio Conte.
"A fourth Old Trafford Premier League draw in a row will be a popular prediction."
Twitter: @Guymowbray
Manchester United manager Jose Mourinho on Chelsea: "It's not for no reason that they are top of the league.
"It's not just because they're fresh but they have individual quality and they have collective quality.
"They have a certain style of play, they stick to it, they do it very well.
"For me, it is just a game. One more game. No difference for me."
Chelsea head coach Antonio Conte: "I have zero problems [with Mourinho]. It's only a sporting competition between him and me.
"There is a game of football. I want to try and win with my team. He wants to try and win with his team. It's normal to be at conflict during the game, but only a sporting conflict."
I was tempted to go for a draw but, because of United's schedule, I am going to back Chelsea to take another step towards the title.
Prediction: 0-2
Lawro's full predictions v singer Sting and his son.
Head-to-head
Manchester United
Chelsea
SAM (Sports Analytics Machine) is a super-computer created by @ProfIanMcHale at the University of Salford that is used to predict the outcome of football matches.
Edinburgh City will play East Stirlingshire in the Scottish League Two play-off after completing an aggregate victory over Cove Rangers.
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Nigerian author Chimamanda Ngozi Adichie has won the US National Critics Book Prize for her novel Americanah.
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Goalkeeper David de Gea is expected to start for Manchester United after missing out against Sunderland with a hip problem and only making the substitutes' bench against Anderlecht.