Milner proved his manager right by stroking the ball in from 12 yards to send the Reds above Manchester City by two points, although the Citizens have played a game less.
Klopp felt vindicated by the conversion and the decision to bring Milner off the substitutes bench for Adam Lallana in the 72nd minute.
Milner was given the chance to score from the spot after Fulham goalkeeper Sergio Rico had felled goalscorer Sadio Mane in the box. The Spanish stopper fumbled a shot from Mohamed Salah then collided with Mane as the Senegal international forward attempted to take the ball around him.
Klopp was satisfied with the decision to award the penalty.
Even so, Jason Burt and Alan Tyers of the Daily Telegraph felt Mane had "made the most of it, going down dramatically."
The winning goal also helped Milner make amends after midfielder's sliced clearance had put Virgil van Dijk under pressure, leading to Fulham's equaliser. Van Dijk uncharacteristically misjudged the clearance, allowing former Reds winger Ryan Babel to tap in.
Milner explained his gaffe to Sky Sports (h/t Neil Jones of the Liverpool Echo):
His mistake may have been costly had it not been for Milner's scoring touch, which helped maintain an incredible record:
The 33-year-old once again proved something of a lucky charm on a day when Liverpool needed a slice of fortune to regain control of the title race. Milner and Co. have the defensive resolve to stay strong, as well as the efficiency in forward areas to win even on days when they don't play well.
It's an ability usually reserved for teams that outlast rivals in tight title races. City's game in hand means the champions will still be favoured to retain the trophy, but Liverpool's formula for success ensures Klopp's team is unlikely to drop many points between now and the end of the season.