Gareth Southgate made 11 changes to the team that won in Hungary by the same scoreline, with Trent Alexander-Arnold starting in midfield and Patrick Bamford handed a full debut in attack.
Jesse Lingard broke the deadlock after 18 minutes but it was not until the final quarter of the game that the Euro 2020 finalists at last pulled clear on the scoresheet, Harry Kane scoring a penalty before Lingard prodded home a clever second.
Bukayo Saka headed in a late fourth from Lingard's cross after England caught Andorra napping with a quick corner.
Lingard, who has only played four minutes for Manchester United this season, was in the mood to impress from the off. He finished with aplomb with his left foot after Saka's cross was nodded into his path, and only the offside flag denied him a quick-fire second when he raced through and lobbed goalkeeper Josep Gomes.
Bamford almost provided a debut assist for Conor Coady, his header down hooked over by the Wolves defender from six yards.
Reece James crashed a fine effort off the crossbar early in the second half but, with Andorra keeping the home side at bay, Southgate turned to Kane, Jack Grealish and Mason Mount off the bench.
That was the trio that finally got England their second goal: Grealish and Mount combined well, the Chelsea man was bundled over in the box by Christian Garcia, and Kane converted from the spot.
Lingard had a quieter second half but made it 3-0 with 12 minutes remaining, a toe-poke from the edge of the box bamboozling goalkeeper Gomes and finding the bottom-right corner.
And it was the United man who delivered a pinpoint cross for Saka to nod in the fourth, the Arsenal winger marking his 20th birthday in style.