Liverpool enjoyed plenty of the ball early on but were given a scare when Armando Broja escaped on the break before being denied by the trailing leg of Alisson.
A failure to heed that warning was punished with just 13 minutes gone as another Southampton counter ended with Redmond cutting in off the left to thrash a curling effort into the top-right corner.
Klopp was incensed by the failure to award a foul on Diogo Jota in the build-up to that goal, but that did not stop his team responding well to the setback.
They were denied an equaliser when Roberto Firmino was flagged offside in heading home a free-kick, but restored parity after 27 minutes as Minamino smashed home a cute Diogo Jota pass at close range.
Premier League: Joel Matip and Minamino with the goals for Liverpool
That prompted a spell of Liverpool dominance, though they were unable to punish some lax Saints defending before a half-time break that saw Joe Gomez withdrawn due to injury. Jota poked the ball inches wide shortly after the restart following great work from Kostas Tsimikas.
The chances kept coming, Harvey Elliott hitting the side-netting, Jota skewing wide, and Firmino sending in a weak shot from distance. As such, it was no real surprise when the goal eventually came and, in fact, neither was the method.
Only Leicester City had conceded more headers than Southampton in the Premier League this term, while Liverpool had scored a league-high 15 goals in that manner and added another when Matip met Tsimikas’ corner to loop one in.
Southampton tried to rally from there, but struggled to see much of the ball as Liverpool ensured City still have work to do on the final day.
Title race to be decided on Sunday
Manchester City must beat Aston Villa at the Etihad Stadium on Sunday to avoid the possibility of any last-day drama. A draw or a defeat would allow the Reds into top spot at the death, should they take victory over Wolves at Anfield.
Liverpool go into that meeting with Wolves on Sunday with a hope of lifting the title. Southampton finish their campaign with a trip to Leicester.