The reign of Nathan Jones got off to a bad start as his new team lost at home.
Roberto Firmino put the disappointment of missing out on a place in the Brazil squad forQatar behind him by heading the Reds ahead before Che Adams equalized.
Darwin Nunez scored two goals in the first half to give the Reds a 2-0 lead.
One of seven Reds players heading for the World Cup as it stands, Nunez made it 2-1 when he poked past the opposing player from close range and tapped home the third after Andy Robertson's assist.
The Reds would have won by more if it weren't for two outstanding stops by the 20-year-old.
Alisson made a great save in the 77th minute to deny Adams a goal.
Ahead of the games to come later on Saturday and on Sunday, the Reds moved up into sixth place, five points off a spot in the European competition.
The Reds were without Ibrahima Konate and Jordan Henderson, who are due to go to the World Cup with France and England respectively, while the manager was banned.
In the absence of their German boss, the Reds scored all three of their goals in the opening 45 minutes.
There are seven players going to the World Cup for the Reds, including Alisson, Fabinho, Konate, Trent Alexander-Arnold and Jordan Henderson.
The next few weeks will be spent with those who are not with him.
After Firmino scored his seventh goal of the season, all of his teammates celebrated with him.
When James Milner came on in the second half for his 600th appearance in the premier league, it was a landmark moment for the 36-year-old.
The upcoming league break will give Jones time to get his ideas across to his players.
The Welshman takes over from Hasenhuttl, who was sacked after last week's home defeat toNewcastle.
The Saints have one win in 10 games in the top flight.
Time will tell if Jones' appointment pays off due to his lack of experience in the top flight.
Jones will be encouraged by the way his players responded after falling behind in the game.
The performance of Bazunu, who was playing in League One on a loan, was positive but this is a young Saints team.
The Saints' line-up against the Reds included two players who are going to the World Cup with Germany, as well as 20-year-old Bazunu.