A group of youngsters climbed over the hoardings around the pitch at full-time and ran towards Andy Robertson to collect his shirt.
Another spectator entered the pitch midway through the second half and was led away by stewards.
Merseyside Police is investigating the matter and UEFA confirmed on Friday it has opened disciplinary proceedings against Liverpool, charging the club with contravening article 16(2) of its disciplinary regulations, which includes a direct reference to pitch invasions.
It is not the first time a Liverpool supporter has run onto the pitch this season, with the most high-profile example being August's penalty shootout triumph over Chelsea in the UEFA Super Cup when a fan slipped and injured goalkeeper Adrian.
Speaking at the time of that incident, Liverpool boss Jurgen Klopp said: "There is no doubt about how much we love our fans but if they could all stop doing that, because that's the worst example I have heard about.
"It happened against Manchester City in the Community Shield when someone was running on the pitch and against Norwich someone ran on the pitch. It's not funny."
Salzburg has also been charged by UEFA over the throwing of objects by supporters during the game, in which Mohamed Salah struck the winner for the UEFA Champions League holder after the Austrian side recovered from three goals down to level at 3-3.
Both cases will be dealt with on 17 October by the UEFA control, ethics and disciplinary body.