The 24-year-old, who has 22 caps for his country, takes over from Scott Brown following his retirement earlier this year.
"It's a massive honour for me and my family – it's a great day for us," Robertson said of his appointment. "I'm looking forward to leading the team out on Friday, and leading them out in the future.
"It's probably the pinnacle of my career. I look forward to trying to lead this country back to major tournaments."
Scotland manager Alex McLeish added: "Andy was the outstanding candidate. His rise-and-rise at Liverpool and at national level was sensational.
"He carries himself well and is a tremendous professional. I know a lot of players inspire each other and Andy is an inspirational figure.
"The story of his career is fairytale stuff – when you consider he's had setbacks a couple of times; it is the epitome of a true Scot to never give up."
Robertson will skipper Scotland for the first time on Friday when they welcome World Cup semi-finalists Belgium for a friendly.