Victory ensured Klopp's side advanced into the knockout phase as group winners and means the Premier League has five clubs in the last 16 of the competition for the first time.
The score was nearly doubled a minute later thanks to a classic counter-press, as Salah fed Roberto Firmino with a lovely back-heel, but the Brazilian's unsafe cross was cleared.
Sevilla are now second in Group E and visit Maribor, though they will have their eyes peeled on the events at Liverpool. Coutinho completing a hat-trick through perhaps a fortunate deflection from a Spartak defender.
Liverpool was one of four English teams to top a group, along with Manchester City, Manchester United and Tottenham.
"Wow. I have nothing to say about this", said Klopp, whose team sailed through to the Champions League knockout stage.
"They've got people with pace, they all score goals". He now also holds the distinction of being the club's fastest-ever Champions League scorer at Anfield when, inside three minutes, he provided some early breathing space after Georgi Dzhikiya pulled down Salah as he attempted to meet the Brazilian's cross.
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Senol Gunes' side won't be easy to beat as they embark on their first foray into the last 16 in 31 years, especially with a second leg in front of their vociferous home fans.
Prior to this game there were still doubts as to whether Liverpool had what it takes to qualify.
It is also why Coutinho's value remains innumerable to Liverpool, with a direct hand in 15 goals from his last 11 appearances. Milner delivered an inviting cross for Mane to thump home on the volley less than two minutes into the second half. It's half man of the match poll and half power ranking, and we want to use it to find out who you think played well after each Liverpool match and, at the other end, who you'd maybe like to see given the next game off.
Spartak finish third in the group, behind second-placed Sevilla, and move into the Europa League.
Liverpool have now scored the most Champions League group-stage goals of any English team in history.