Hastings was forced off in the first half of the game after colliding with Scarlets number eight Vaea Fifita.
In a statement, Glasgow Warriors said: "After a short spell in hospital after suffering a facial fracture, [Hastings] has now been released and is back home - he will now begin working with our medical team on his return to play pathway."
Hastings has had terrible luck with injury over the last two years but had been showing good form for Glasgow and played well in Scotland's November games.
Speaking ahead of Glasgow's Champions Cup meeting with Sale Sharks, defence coach Peter Murchie confirmed Hastings had undergone surgery and the recovery time would be months rather than weeks, a blow both for his club but also for the national team.
Murchie said in The Scotsman: "It's a long recovery process so we wish him all the best. We'll look after him and he'll get the best of care. Obviously, it was pretty clear that it was a nasty one straight away. Initially we didn't fully realise that it was the jaw but it's a huge contact.
"It must be incredibly frustrating for him. That's been a couple of years now when he's had some really bad luck. He knows how to recover and I'm sure he'll be back. But everyone's gutted for him. He was starting to put together some really nice performances so it's incredibly frustrating for him."