Roberto Firmino's injury-time strike sealed a deserved win in their opening group game after a two-goal lead was earlier cancelled out.
He simply has not looked as sharp as he did previous year and the game against PSG may have brought an answer as to why.
But Firmino, who came on as a second-half substitute after recovering from his eye injury, scored a superb victor in injury time to seal three points for Jurgen Klopp's side.
The Belgian scored a quality goal in the first half which meant that Liverpool went into the break with the Paris-based club breathing down their necks. I am happy with my team's performance - the match was of a high level.
Mbappe looked like he had snatched a point for PSG on 83 minutes, pulling it back to 2-2 after the French champions had trailed 2-0. Conceding in the first minute of stoppage time at the end of an electrifying encounter at Liverpool is just what happened when Tuchel's Borussia Dortmund was beaten 4-3 by Klopp's side in the Europa League quarterfinals in 2016.More news: Valencia vs. Juventus - Football Match Report
"A few times, he has lost his head when we haven't played well, but he'll surprise you: there are times when things aren't going badly and he still loses it, and times when you think he's going to, but he doesn't".
Silva was in the right place to clear as Alexander-Arnold crossed, and with 20 minutes remaining Klopp made a change, with Firmino coming on in place of scorer Sturridge.
"This team is built for winning the Champions League".
"It was good, really good in all departments pretty much", Klopp said.
Liverpool's Champions League win over Paris Saint-Germain proved they are on a par with Europe's best, according to Joe Gomez.
"The three in midfield also (were great) but they can only do that when the front three play like they do and in that number nine position he (Sturridge) did".