Ascot United and Bracknell Town were both knocked out by Cricklade Town and Witney Town respectively.
Nicky Bateman put the Satsumas ahead from the penalty spot and further goals from Luke Lee and Bateman secured their passage to the next round. Kieran Davis and Sean Flaherty scored for Hinksey.
Meanwhile, the Yellamen had three players sent off as they suffered a 4-2 extra-time defeat to Cricklade Town.
The scores were locked at 2-2 after 90 minutes with Rob Saunders and Ian Brooks netting for Ascot.
But the Yellamen were clearly hampered by the dismissals of Jon Bennett, Liam Woozley and Saunders, and the lack of personnel told in extra time with Cricklade scoring twice more to come away with the victory.
The Cricklade goalscorers were Jack Allan (2),Ryan Thomas and Ben Carter.
Bracknell Town fought back from two goals down but were narrowly beaten 3-2 by Hellenic Premier Division opponents Witney Town.
Conor McDonagh's double gave Witney a two-goal cushion but Scott Williams and Shane Cooper-Clark levelled the scores for the Robins before James Dobson scored the winner shortly after.
Binfield slipped to sixth in the Hellenic Premier Division after conceding a late goal to go down 3-2 at home to Oxford City Nomads.
Carl Davies put the Moles ahead twice and they were leading 2-1 with 12 minutes to go until the wheels came off and Adam Lovegrove scored the winner for Oxford two minutes from time.
Finchampstead netted eight without reply against Milton Utd in Division One East.
Leading the way with four goals was Harry Swabey with further goals coming from Greg Alexander (2), Josh Sapsed and Stewart Manicom.
Sandhurst were thrashed 6-1 at home to Molesey in the Combined Counties Premier Division while Eversley & California narrowly lost 3-2 to AFC Croydon Athletic in the second round of the EL Records Challenge Cup.
* SEE THURSDAY'S NEWS FOR FULL REPORTS AND REACTION.