But they did at least keep themselves in the running with a spirited comeback at St Mary's after falling behind to goals from Dusan Tadic and debutant Jan Bednarek.
Liverpool will face Roma in the semifinal of the UCL, after dumping out City and Barcelona respectively.
Salah's contribution in the Champions League perhaps the icing on the cake for the Reds as the 25 year old consolidated his position in the team with eight strikes in 10 Champions League matches including a valuable one in the second leg of the quarterfinal against Manchester City. "I must stress that we have an outstanding scouting department". If any German goes to Bern in Switzerland he cannot avoid thinking about 1954. You think, 'Ok, it happened here, well done, all these guys'.
Salah certainly still had motivation in his battle to beat Kane to the Golden Boot, but there was a sense it might not be his day when he failed to win a penalty after a coming together with Nathan Ake.
'Those Liverpool teams were great but they became great in these situations and now people say "They did it like this".
Andy Cole, Alan Shearer (3), Kevin Phillips, Thierry Henry, Cristiano Ronaldo, Robin van Persie and Luis Suarez are the others to have achieved it.
Referee Chris Kavanagh was unmoved, but that ball over the top to Salah showed the way to get him in and he finally scored in the 69th.
"Kenny Dalglish is over the moon and completely on fire". When it's like constantly so fantastic, it's the first step to not be fantastic anymore and we have to stay this kind of angry.
Yet even that attack was halted by the assistant's flag as Ryan Fraser was offside prior to collecting Charlie Daniels' back-heel and crossing for Jermain Defoe to tap home.
He said: 'I will watch them for sure because it is destiny.
For all of Liverpool's dominance, the hosts had only one goal to show for it in the first half as Salah continued to be frustrated, with Steve Cook's brilliant challenge denying him.