Babos, who had a hand in the Rams' opener for Josh Shonibare - the first goal Villa have conceded in the competition, hit a superb finish to double his side's lead.
Villa substitute Tyreik Wright pulled one back for the home side but Derby held on through a nervy final 10 minutes to secure the victory and their place in the top two.
There is just one game remaining in the group stages.
Derby County: Barnes, Gordon, Mcdonald, Bird, Bateman, Hunt, Mitchell-Lawson, Sibley (Wilson, 69), Cresswell (Buffonge, 61), Babos, Shonibare. Subs not used: Eyoma, Yates, Karic.
Aston Villa: Sinisalo, Walker (Burton, 79), Rowe, Bazeley-Graham, Revan, Prosser, Knibbs, Vassilev (Brunt, 63), Mooney, Clarke, Archer (Wright, 55). Subs not used: Patterson, Lomax.