Archer, who was representing Manchester United, got off the mark first despite a strong first-half performance by the Newcastle United man.
And the Barbados-born bowler looked to be going into the break with a 1-0 lead only for Aarons to strike just seconds before the half-time whistle as Dwight Gayle finished off a slick passing move.
Despite creating a number of chances after the interval, Aarons fell behind again with a stroke of bad luck as a loose ball fell to Nemanja Matić inside the box and it was duly tucked beyond Martin Dúbravka.
Aarons went close to an equaliser through Isaac Hayden with around 20 minutes to play, only for his shot to roll wide.
But just seconds later, Archer struck again as Marcus Rashford got forward to slot in at the near post before increasing his advantage again through Anthony Martial.
Archer progresses to play Sheffield United's John Egan in the Last 16.
Watch LIVE
All remaining matches will be streamed live on the Sky Sports YouTube and Twitch channels, as well as BBC Sport website and iPlayer.
Finals Day, which takes place on Saturday 9th April, will also be shown live in the UK on Sky Sports' Premier League and Main Event channels. The tournament is also being televised around the world by the Premier League's international broadcast partners.