The full-back suffered the injury during Sunday's 2-1 home defeat at Anfield and Liverpool confirmed he will now undergo reconstructive surgery.
The club offered no timeframe for Kelly's recovery but cruciate injuries customarily mean a spell of at least six months out.
Kelly tweeting: "It's not the worst injury I've had, and for sure I'll be fighting for my place next year when I'm back fitter and stronger!"
Meanwhile, Liverpool said fellow defender Daniel Agger suffered a "significant bone bruise" on his left knee but had not sustained ligament damage.
Agger was stretchered off in the second half of Sunday's game.
Forward Fabio Borini will undergo an MRI scan on his right ankle, which has "considerable swelling" but no break after yet another incident at Anfield.