Focusing on his studies, Robertson was on the brink of starting university at undergraduate degree level before he was called up to the Queen’s Park first team at the beginning of the 2012–13 season. He made his debut for the club in the Scottish Challenge Cup in a penalty shoot out win against Berwick Rangers, and went on to make a total of forty appearances during the season, helping the club to a third-place finish in the 2012–13 Scottish Third Division. His first goal for the club came in a 2–1 defeat to East Stirlingshire on 13 November 2012.

Robertson signed for Scottish Premiership side Dundee United, along with Queen’s Park teammate Aidan Connolly, on 3 June 2013; he took the squad place of the departing Barry Douglas who had made the same move three years earlier. Having been ignored in their requests for a transfer fee for Douglas, Queen’s Park (an amateur club) protested at the prospect of more players departing to United for nothing and eventually agreed a deal involving a percentage of future transfers, which would later prove beneficial.

Robertson immediately became part of manager Jackie McNamara’s first team plans, making his debut for the club on the opening day of the season in a 0–0 draw with Partick Thistle. On 22 September 2013, Robertson scored his first goal for Dundee United, in a 2–2 draw against Motherwell. He ran from his own half before sending a low left-foot drive in to the goal from 22 yards. Soon afterwards he agreed a new contract with United, until May 2016. He was voted SPFL young player of the month for September 2013 and player of the month for November 2013. On 12 April 2014, Robertson played in the Scottish Cup Semi-final against Rangers at Ibrox, which United won 3–1. In April 2014, Robertson won the PFA Scotland Young Player of the Year award and was also named in the PFA Scotland Team of the Year for the 2013–14 Scottish Premiership.

Robertson was first selected by the Scotland under-21 team in October 2013. He made his debut as a substitute in a 2–1 win against Slovakia. Robertson was selected for the under-21 team in November 2015, as the full national team did not have a fixture.

In July 2014, Dundee United accepted an offer of £2.85 million for Robertson from English Premier League club Hull City. The transfer was completed on 29 July, as Robertson signed a three-year contract with Hull. As Stan Ternent, Hull City’s chief scout explained, “I’d been looking at Stuart Armstrong but [Robertson] was the no brainer … he’d a history with Celtic and he was always a determined lad given how he’d recovered from his setbacks. You could see straight away he had ability and he can only get better.”

Robertson was first named in the full Scotland squad for a friendly against Poland on 5 March 2014. He came on as a substitute in the second half, as Scotland won 1–0 in Warsaw. Scotland manager Gordon Strachan said afterwards “Andy came on there and the first time he picked it up he drove about 30 yards. I thought ‘that’s fantastic’. Absolutely no grey area, I’m going to do what I do. I loved seeing that first touch.” Robertson made his first start for the national team in a 2–2 draw with Nigeria on 28 May 2014.

Robertson scored his first international goal in a friendly against England at Celtic Park in November 2014, scoring Scotland’s only goal as they were beaten 3–1.

Despite the release of several other first-team players, Robertson opted to stay at City. His first goal for the club came on 3 November 2015 away to Brentford where he opened the scoring in a 2–0 win to put Hull top of the Championship table on goal difference. He started the 2016 Championship play-off Final against Sheffield Wednesday, which Hull won 1–0 to secure promotion to the Premier League. The team spent just one season back in the top flight, however, before being relegated yet again.

On 21 July 2017, Robertson signed a long-term deal with Liverpool for an initial fee of £8 million. On 19 August, he made his debut in a 1–0 win over Crystal Palace, winning the Man of the Match award. Robertson started the 2017–18 season as understudy to Alberto Moreno, but got a run of games when the Spaniard was injured in December. His performance in the 4–3 victory against league leaders Manchester City on 14 January earned praise from the Liverpool supporters. He scored his first goal for the club on the final match day of the 2017–18 season against Brighton & Hove Albion in a 4–0 victory.

Robertson is engaged to Rachel Roberts, who gave birth to their son, Rocco, on 26 August 2017. In January 2019, the couple welcomed their second child and first daughter, Aria.

On 3 September 2018, Robertson was appointed as Scotland captain by manager Alex McLeish. After a 2–1 defeat against Israel in October 2018, Robertson said that both he and fellow left-back Kieran Tierney were being played out of position in the 3–5–2 system adopted by Alex McLeish to accommodate both players.

In March 2018, Robertson donated a signed Liverpool shirt of his teammate Roberto Firmino to a young boy who had given his pocket money to a local food bank. The Sunday Times reported in March 2020 that Robertson had donated significant sums to food banks in the Glasgow area.

Robertson continued to play regularly for Liverpool during the 2018–19 season when Liverpool were runners-up in the Premier League. He signed a new contract with the club in January 2019, which is due to run until 2024. He has been hailed as the best left back in the Premier League by Kevin Kilbane and the best left back in the world by Phil Neville. On 25 April, he was named in the PFA Team of the Year alongside Liverpool teammates, Trent Alexander-Arnold, Sadio Mané and Virgil van Dijk.

On 1 June 2019, Robertson played the full 90 minutes of the Champions League final as Liverpool beat Tottenham Hotspur to secure their sixth European Cup. He became the first Scot to win the tournament since Darren Fletcher (an unused substitute in 2008) and the first to actually be on the field in a winning team since Paul Lambert in 1997.

In the European champions’ first home game of the Champions League for the 2019–20 campaign on 2 October, Robertson scored his first European goal in a dramatic 4–3 victory against Red Bull Salzburg. On 2 November 2019, Robertson scored his second Premier League goal for Liverpool, equalising in the 87th minute against Aston Villa, with Liverpool eventually winning 2–1 in stoppage time.

In a UEFA Euro 2020 qualifier against Cyprus on 8 June 2019, Robertson scored the opening goal of the game with a long-range strike. Scotland eventually won 2–1. In November, he was one of five Scotland players to withdraw from the national squad due to injury.

In 2020, he and fellow Liverpool full-back Trent Alexander-Arnold starred in their own digital series on IGTV called ‘Wingmen’.