The Greeks went ahead thanks to a cool finish from Moroccan striker El Kaabi on 16 minutes and he made it 2-0 on half an hour as Villa's defence fell apart on Thursday.
The Birmingham-based outfit levelled through top scorer Ollie Watkins on the stroke of half-time after a neat move and managed to equalise after the break when Moussa Diaby scored from a tight angle.
But having got themselves back in it, Villa then conceded a penalty through Douglas Luiz, who later missed one himself, and up stepped El Kaabi to complete a famous hat-trick. Santiago Hezze then added a deflected fourth on 67.
The game was delayed due a medical emergency in the crowd.
Unai Emery, who has won this competition four times, now faces a huge challenge as his Villa side look to qualify for their first European final since winning the European Cup in 1982.
Record Greek champions Olympiacos will meanwhile be eyeing their maiden European final when they welcome Villa to Piraeus next Thursday.
In Florence, Riccardo Sottil's effort from distance had La Viola in the lead but Brugge soon hit back from the spot through Hans Vanaken after video review showed Cristiano Biraghi had committed a foul.
Fiorentina were back in front on 37 minutes through forward Andrea Belotti only for Thiago to level on the break in the second half after seeing teammate Raphael Onyedika sent off moments earlier for a second booking.
But Fiorentina are no favourites for the second leg in Belgium on Wednesday thanks to M'Bala Nzola's stoppage-time winner.
