Cash gave Aston Villa an early with his first goal since May 2022, before the full-back added a second in the 20th minute to put the visitors in the driving seat.
After experiencing a difficult first period, Burnley gained a foothold within two minutes of a restart when Lyle Foster netted his first Premier League goal.
However, Burnley's hopes of claiming a positive result were effectively ended just after the hour-mark, as Moussa Diaby restored Villa's two-goal advantage.
Unai Emery's side managed to hold firm to round off a successful week, where they scored 12 goals in three consecutive victories across all competitions.
Fresh from thrashing Hibernian 5-0 in Thursday's Europa Conference League clash, a confident Aston Villa side flew out of the gates against Burnley.
The visitors hit the front in the eighth minute when Ollie Watkins lofted the ball over the onrushing James Trafford, giving Cash the simple task of tucking home from close range.
Cash had only scored two goals in a game on just one previous occasion, scoring a brace as a Nottingham Forest player in a 3-3 draw against Norwich City in December 2018, but that statistic quickly changed at Turf Moor.
Following a slick passing move, Cash calmly swept the ball in at the near post to double Aston Villa's lead inside the opening 20 minutes.
A dominant first-half performance was nearly capped off in the closing stages when Watkins went close to adding a third, but Trafford made a fine stop to keep Burnley within touching distance.
The half-time break appeared to work wonders for the hosts, who pulled a goal back through Foster in the 47th minute.
Foster utilised his strength to hold off Pau Torres, before quickly swivelling to fire the ball past the despairing Robin Olsen.
Having gained a foothold, Burnley's search for a leveller nearly proved fruitful when a corner from the left flank found the towering Hannes Delcroix, but Olsen was on hand to deny the Belgium defender.
Just two minutes later, Diaby restored the visitors' two-goal cushion, striking the ball low into the bottom corner from Lucas Digne's neat cut-back.
While Jay Rodriguez went close to finding the net in the closing stages, Aston Villa saw out the remainder of the contest with minimal fuss, ensuring they could celebrate their first away win of the Premier League season.
The win moves Aston Villa into sixth position in the Premier League table, while Vincent Kompany's side are languishing in the relegation zone after losing their opening two games.