Liverpool were outstanding favourites going into the tie which ended with a fantastic performance from the Swans.
Scott Tannock scored the opener when he met Josh Pike’s free-kick just before half-time.
Only minutes later, James Loveridge doubled the scoring with a sweet volley.
Jandir Zola netted a bizarre third for the Swans and a stoppage time consolation goal by Lewis Hatch was not enough to retrieve any hope for Liverpool.
Elsewhere in the FA Youth Cup, West Ham United beat Wolverhampton Wanderers away from home 3-2.
Nottingham Forest edged past Sunderland, winning 2-1 and West Brom beat Barnsley 3-0.
There is still a range of FA Youth Cup third round games left to play in the coming week.
FA Youth Cup Third Round Results
Newcastle United 3 – 0 Darlington
Oxford United 1 – 2 Bolton Wanderers
West Bromwich Albion 3 – 0 Barnsley