Buenos Aires in Argentina and Medellin, Colombia, are the other potential hosts announced by International Olympic Committee (IOC) president Jacques Rogge in Switzerland today.
The Scottish bid was highlighted by the organising committee for its use of existing venues and experience of hosting international sporting events.
The promotion of the event as a legacy of the 2012 Olympic Games and the 2014 Commonwealth Games was also identified as a positive factor.
The host city will be decided in July.
Lord Coe, chairman of the British Olympic Association (BOA), said: "It is an honour to have Glasgow shortlisted to host the Youth Olympic Games (YOG) in 2018.
"London 2012 inspired young people like never before and there is now a once-in-a-lifetime opportunity for the UK to take the next step on the Olympic journey to empower young people globally.
"Glasgow cannot only stage a compact, sustainable and financially sound Youth Olympic Games, but take a unique journey in history hand-in-hand with the Olympic movement fuelled by the passion and warmth of Scotland.
"A Youth Olympic Games in Glasgow would empower young people worldwide to believe they can be champions in their lives and, at the same time, inspire the world to support them."
Up to 3,600 athletes aged between 15 and 18, are set to take part in 28 sports at the 12-day 2018 Games.
The YOG committee found that Glasgow's budget was sound and that the bid was "clear and good quality".
Guadalajara in Mexico and Rotterdam in Holland were excluded from the shortlist due to issues with financial guarantees.
Britain's most successful female rower Katherine Grainger believes Glasgow would be the "perfect" host.
The Olympic gold winner said: "For many young athletes the Youth Olympic Games will be the first step towards competing at the very highest level and it is fantastic to think that they may be taking those steps in Glasgow.
"The city has a proven track record of delivering major international sporting events in world-class facilities in front of passionate crowds - something that any young athlete would respond to."
The bid is being run by the Scottish Government, Glasgow City Council and the BOA.
Scottish sport minister Shona Robison said: "Having the opportunity to host the Youth Olympic Games is hugely exciting for Scotland.
"Glasgow 2018 will put the aspirations and talents of young people at the centre of the Games, and my congratulations go to the young people at the heart of the bid."
Glasgow City Council leader Gordon Matheson said: "Glasgow is extremely proud to have been granted the privilege of being shortlisted by the IOC."
Prime Minister David Cameron visited Glasgow last year to give his backing to the bid and was delighted to see it shortlisted.
"This is fantastic news for Glasgow and a massive vote of confidence in the UK's ability to host major sporting events," Mr Cameron said.
"Young people are an important part of the legacy of the 2012 Olympic and Paralympic Games, and I hope that competing for the bid will open their eyes to the possibility of sport and inspire the next generation of Olympic heroes.
"When I visited Glasgow last year to back the bid, I was impressed by the commitment and dedication of the team who are working so hard to win the bid, and I wish them every success."
The regeneration of an area of Glasgow is being brought forward as part of the bid.
Sighthill, in the north of city, will be home to the athletes' village if the bid is successful, but building work will begin before any host city is chosen because of the timescale of the bid.
Glasgow 2014 chairman Lord Smith said: "Scotland has a growing reputation for staging successful world-class events and securing the 2018 Youth Olympic Games would be a fantastic opportunity to build on the experience and skills developed through delivering an outstanding Commonwealth Games in Glasgow in 2014."
The first summer Youth Olympics was held in Singapore in 2010 and the event will be held in Nanjing, China, next year.
Scottish Conservative leader Ruth Davidson said: "It is tremendous news that Glasgow is now within touching distance of hosting the 2018 Summer Youth Olympic Games.
"With preparations moving apace for next year's Commonwealth Games, the city has put itself in a strong position in offering a world-class bid.
"Glasgow is one of the world's great sporting cities and winning the right to host this prestigious event would cement that well-earned reputation, as well as creating a wonderful legacy.
"Hosting the Youth Olympics would give all our young athletes the motivation to become the stars of tomorrow.
"Let's hope we can now get over the finishing line when the final decision is announced in July."
Labour MSP Patricia Ferguson also welcomed the naming of Glasgow on the three-city shortlist.
She said: "Here's hoping that everyone involved puts forward the best possible case to bring the Games to Glasgow and we manage to hold off the competition from South America.
"We will all be hoping that Glasgow crosses the line first and that we have another reason to celebrate sporting achievement in our biggest city."