The on-loan Chelsea defender rose highest in the 58th minute to head home the goal which ended a five-match winless run.
Paul Caddis hit the crossbar for the hosts and Danny Graham missed a gilt-edged chance as Rovers exited at the third-round stage for the first time since 2014.
The first half was a dour affair but Caddis almost lit it up early on with a superb strike that cannoned off the crossbar.
The Tigers took the lead in the 58th minute when Jon Toral’s deep corner was met by Aina, whose header somehow evaded Jason Leutwiler in the Rovers goal.
Jarrod Bowen almost scored a carbon copy goal a minute later but hit the side netting.
Graham should have equalised but missed the target with only goalkeeper David Marshall to beat, and Derrick Williams prodded over as Blackburn’s 18-match unbeaten run was ended.
Hibernian head coach Neil Lennon claimed referee John Beaton failed to protect his players from 'disgraceful' challenges during a 1-1 draw with St Johnstone.
St Johnstone manager Tommy Wright has accused Neil McCann of deflection tactics as he denied reporting the Dundee boss to the Scottish Football Association.