Questions about his club's chances of finishing in the top four particularly irked the German. The defeat kept the Reds in ninth place and seven points off Manchester City in fourth.
Philippe Coutinho and Daniel Sturridge gave Klopp's men a comfortable advantage after just over 20 minutes. But the Saints fired back with three second-half goals, including a brace from Sadio Mane, to take the points:
When asked during a post-match interview with Sky Sports about his side's prospects for securing UEFA Champions League football for next season, Klopp became irritated: "I don't know about top four. I would say all of you should start more fantasy with your questions because it's always 'yes top four, or no top four.' We are not closer. That is clear. It's not the best position."
When pressed, Klopp called on the interviewer to "count the points." He continued: "Nine games. Twenty-seven [points]. If possible...then yes."
Perhaps the more relevant question was whether Liverpool's recent exertions in the UEFA Europa League, in which the Anfield club dispatched bitter rivals Manchester United to earn a quarter-final berth, had sapped the energy of Klopp's players.
While he tried to play that thought down, the German was keen to point out how Liverpool's opponents have featured less this season: "When you count all the games we have had and all the games Southampton have had, I think at the end of the season, we have had 20 more or something like that. How could I blame my players that they are not in the same shape [in the] second half?"
Instead, Klopp focused his ire on the missed chances he felt prevented Liverpool from killing the game off when they were in front: "You have to finish the game in the first half or the start of second half. We have to accept it was Southampton's day. They played well in the second half, and we didn't. We didn't get enough goals in the first half, so it's our fault."
The reference to missed chances is an interesting one. It speaks to the fact Liverpool's attackers still haven't completely adapted to Klopp's brand of fluid, combination play up front.
Should a striker be Klopp's transfer priority this summer?
His squad still seems short of a pacy and truly prolific centre-forward who can take over games.
More than anything else, the absence of one is likely to cost Liverpool fourth place and a return to Europe's premier club competition via the Premier League.