A chaotic 10-minute spell in the first-half saw both sides score twice before May completed her hat-trick just before the break to give City the advantage.
The Hoops then added a fourth soon after half-time and, despite a late rally from the visitors, secured their first victory of pre-season.
The match started in a stop-start fashion with injuries causing problems for Oxford, Chloe Duckett stayed down for about three minutes early on before limping off and Emma Wilcocks also had her injury concerns.
This made the result and performance even more impressive for City who only had a 13-man squad for the game which meant players had to play out of position and most had to play the full 90-minutes.
The first chance of the match came for Swindon after 15 minutes with Emily Taylor denying their striker from close range.
City then had their first attempt midway through the first-half as Jess Doyle beat two defenders before shooting straight at the keeper.
Then came the remarkable period of play with four goals within 10 minutes of each other. The first was the Oxford opener following good pressure from Danielle Anderson which resulted in the keeper playing the ball straight to her, she then picked out May in the centre of the box who finished into a largely empty net.
Swindon levelled up just five minutes later following a nice through-ball into the box which found their striker who rode a City sliding challenge and produced an excellent finish from a tight angle.
There was an immediate reaction from the Hoops as they won a penalty less than a minute later after May was upended in the box.
The City number 10 then stepped up herself and slotted the ball to the keepers left who couldn't get down to stop it.
This time it was Swindon with an instant response as they drew level again with a nice finish past Emily Taylor.
The visitors almost took the lead themselves soon after as a cutback from the byline was scrambled away by the City defence. Swindon then hit the back of the net from the rebound but the whistle had already been blown for a City free-kick.
May then completed her first-half hat-trick moments after Anderson hit the post with an effort from just outside the box. May then collected the ball, beat a couple of Town defenders before squeezing the ball home at the near post.
The fourth came just three minutes after the break as great persistence in the box resulted in her getting a shot away with her left foot which was too powerful for the keeper to stop.
City then controlled the next 20 minutes with few chances being created by either side, largely due to more minor injuries for both sides.
Swindon then thought they pulled one back following a scramble in the box only to see the assistant had his flag up, much to the confusion of the Town players.
Ellie White then had a great chance after a brilliant run from midfield saw her face the keeper 1-on-1 but her effort was kept out.
Swindon did eventually pull one back in the 90th minute following a mistake from Taylor in the City goal who played the ball straight to their striker who halved the deficit.
They then had a golden chance to equalise in added time after a mix-up at the back led the ball being played to their number 12 who had an empty net to aim at but fired wide from distance.
Oxford held on for an impressive result which first-team coach Waheed Ramzan will look to build on ahead of the upcoming season.