Beavon smashed home just before the half-hour mark to inflict a second successive defeat on Swindon, following their loss at Barnsley last weekend.
The striker punished hesitancy among the Swindon defence with a clinical volley, the Brewers' first ever goal at the County Ground.
Wes Thomas dragged wide from inside the penalty area for the home side.
Jon Obika and Louis Thompson also missed the target while Yaser Kasim had a free-kick pushed over by Jon McLaughlin as Burton held out for the win.
Burton Albion boss Jimmy Floyd Hasselbaink told BBC Radio Derby:
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"I'm very happy. I think when you're coming to Swindon, you know what they're about. They can keep the ball for fun and they can test your nerves and get overloads in certain areas, but we made it very uncomfortable for them. We didn't let them get into their stride, we didn't let them play their game and get into their rhythm.
"For us, it was all about winning the ball back in their half and trying to penetrate in the last third and I think we did that really well.
"We should have scored more goals and that's something we need to work at, but I must say I'm extremely happy with how the team has worked today. We were very solid and we were quite creative with the ball at times, but the belief needs to become greater because they can do a lot more than they think."

Works include a pedestrian and cycle bridge across the A63 and extensions to the city's art gallery and theatre.
The bridge will cost £11.5m and will link the city centre with Hull's waterfront, which is being redeveloped.
The Ferens Art Gallery will get a £4.5m facelift ahead of it hosting the Turner Prize in 2017.
Labour councillor Sean Chaytor, chairman of Hull City Council's planning committee, said the bridge was "essential in working towards improving connectivity between two core areas of the city".
"We hope that the work to start the bridge construction can now get under way for early next year in order for it to be ready by Easter 2017," he said.
Hull was awarded the title of City of Culture in November 2013, beating competition from Swansea, Leicester and Dundee.
The year-long event takes place every four years and is awarded by the UK government with the aim of helping tourism and the economy.

The roof of the building on Main Road in Otterbourne, near Eastleigh, collapsed during the blaze, which broke out at about 03:30 GMT.
No injuries were reported at the site, which also contains workshops and offices. An investigation is under way.
The neighbouring convenience store, which was saved, is due to reopen later.
Ty Whitlock, commanding officer at the scene, said: "When we pulled up we were faced with flames coming out of the roof, coming out of the glass doors and windows and obviously huge amounts of smoke coming from the eaves."
Stuart Beavon's first-half strike moved Burton to the top of League One with their first ever win at Swindon.
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A number of building projects planned for Hull's year as UK City of Culture in 2017 have been approved by the council.
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Up to 40 firefighters have been tackling a large fire which destroyed a garage in Hampshire.