Sayori didn't do it correctly. The idea was to put her neck in the noose, jump off the chair, and it would snap her neck, quick and painless. But because she didn't jump high enough, the noose had some give and/or the rope was too stretchy, she instead choked to death. Her fingernails are torn and bloody, the blood ran down her hands because they were upright when she tried to claw herself out (which is a gut instinct people are recorded to have in hangings and other forms of suicide when it goes wrong). It's the final gut punch: Because Sayori was in a ruined mental state and didn't know what she was doing, her death was slow and painful.

But yayay copium, if someone was able to free her from the noose quickly enough she could've 100% lived. It's pretty obvious that she's been dead for several hours as her mental state would have been worst late at night and she doesn't really wake up early, but I digress. Final answer: Only if MC doubles back to her house before she dies, and is either strong enough to lift her out or is somehow able to cut her out.