Not since they squeezed past League Two side Wycombe Wanders in 2016 have Villa been involved in the competition beyond early January.
Cash's goal carried an element of good fortune.
Following a short corner, his shot struck the back of sub Manu Latte Lath, wrongfooting the previously unbeatable Tom Glover in the Boro goal.
The Championship side had their moments but the visitors deserved this hard-fought win.
Ezri Konsa hit the woodwork, Alex Moreno had a decent penalty shout turned down and the visitors were denied on several occasions by Glover.
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