Fulham joined Sheffield United and West Bromwich Albion in going down to the Championship, wrapping up the Premier League's relegation battle.
Leeds had lost all five of its games in London this season, and was winless in its last 16 visits to the capital in all competitions, losing 14.
Klopp is also now under the spotlight as his decision to make seven changes from Thursday's 1-0 defeat to Chelsea did little to aid his side's confidence.
Pogba created space for himself just outside the corner of the penalty area and bent a powerful shot past Fulham goalkeeper Alphonse Areola and into the far corner of the net in the 65th minute.
Fulham says Scott Parker returned a negative test but has been self-isolating this week in accordance with government guidelines.
Liverpool drawing their game against Fulham resulted in the Reds staying at the second position in the Premier League standings.
Fulham has invested heavily in its playing squad, only to start without a point from two games before its first league win on Sunday.
Seri was swayed by the promise of regular first-team football and Chelsea's case was not helped by being still in a state of flux over the status of manager Antonio Conte.