The Northern Irish club led 1-0 at half-time and although B36 hit back just after the break, Linfield progressed after the 1-1 draw.
Former Northern Ireland international Feeney was appointed to his first managerial role with the club in April and oversaw a 2-1 win over B36 in his first game last week.
Having gained an important win in the Faroe Islands, Michael Carvill gave the home side an ideal start at Mourneview Park.
Carvill struck after 17 minutes following an impressive knockdown from Aaron Burns to edge his side closer to the second qualifying round of the UEFA Europa League.
Linfield made a dreadful start to the second half, however, conceding an early penalty as goalkeeper Jonny Tuffey brought down Adeshina Lawal in the area.
Lawal equalised from the spot but the Faroese visitors were unable to add a second.
Instead, Feeney's men held off late pressure from their opponents to book a meeting with Sweden's AIK Stockholm in the next round.
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
