The 16-year-old slotted in Sone Aluko's backheel after the break and then fired home just before the hour mark.
Tom Cairney had put Fulham ahead with a 25-yard strike into the top corner.
After Daryl Murphy curled in a consolation for the hosts, Tim Ream missed a chance to make it 4-1 when he sent a stoppage-time penalty wide.
Sessegnon appeared to want to take the spot-kick to complete his hat-trick after Paul Dummett brought him down, but USA international Ream took the ball and smashed it past the right post.
Fulham have now won five of their last seven league games and are just two points behind sixth-placed Sheffield Wednesday with a game in hand.
They thoroughly outclassed an out-of-sorts Newcastle side, with skipper Cairney's second fantastic long-range goal in the space of a few days, after his stoppage-time equaliser against Leeds on Tuesday, setting the tone.
However, it was left-back Sessegnon, reportedly being scouted by several Premier League clubs, who provided two moments of composure to seal three points for his side with his fifth and sixth goals of the campaign.
It was a nightmare afternoon for Rafael Benitez's Magpies, a year to the day since the Spaniard took charge, with their advantage over third-placed Huddersfield now down to six points.
But over the past 10 seasons in the Championship, an average of 86 points has been enough to gain automatic promotion, so three more wins could be enough to earn an immediate return to the Premier League.
Newcastle manager Rafael Benitez: "We knew it could be a difficult game, and we knew we had to do well at the beginning and not make mistakes. But we started making mistakes, and we paid for that.
"Fulham are a good team - they passed the ball well and they have pace. We were giving the ball away, and that was giving them the chance to play in the way that they like to play.
"When you have tried and have made mistakes, you have to analyse why. It is not as though the players were not trying in terms of effort. Sometimes, it is just the wrong decision, and that is something we have to improve."
Fulham boss Slavisa Jokanovic: "I believe at the end my team deserved the result. It is a great step for us, everything is open, although some teams have more points than what we have right now.
"I believe Newcastle will be a Premier League team. They play a different style of football - they are physical and a very strong team.
"We are not in a position to think about being a promoted team, but we are going to push hard for the top six. If you have enough strength for this then we have a 25% chance to be a Premier League team next season."
Match ends, Newcastle United 1, Fulham 3.
Second Half ends, Newcastle United 1, Fulham 3.
Foul by Ryan Sessegnon (Fulham).
Matt Ritchie (Newcastle United) wins a free kick on the right wing.
Scott Malone (Fulham) is shown the yellow card.
Foul by Scott Malone (Fulham).
Matt Ritchie (Newcastle United) wins a free kick on the right wing.
Substitution, Fulham. Scott Parker replaces Stefan Johansen.
Penalty missed! Bad penalty by Tim Ream (Fulham) left footed shot is close, but misses to the right. Tim Ream should be disappointed.
Penalty Fulham. Ryan Sessegnon draws a foul in the penalty area.
Penalty conceded by Paul Dummett (Newcastle United) after a foul in the penalty area.
Paul Dummett (Newcastle United) is shown the yellow card for a bad foul.
Tom Cairney (Fulham) wins a free kick on the right wing.
Foul by Paul Dummett (Newcastle United).
Substitution, Fulham. Cyriac replaces Sone Aluko.
Attempt blocked. Jamaal Lascelles (Newcastle United) right footed shot from outside the box is blocked. Assisted by Jonjo Shelvey.
Substitution, Fulham. Ryan Fredericks replaces Neeskens Kebano.
Foul by Denis Odoi (Fulham).
Christian Atsu (Newcastle United) wins a free kick in the defensive half.
Sone Aluko (Fulham) wins a free kick on the right wing.
Foul by Jack Colback (Newcastle United).
Offside, Newcastle United. Daryl Murphy tries a through ball, but Yoan Gouffran is caught offside.
Hand ball by Christian Atsu (Newcastle United).
Jonjo Shelvey (Newcastle United) is shown the yellow card for a bad foul.
Tom Cairney (Fulham) wins a free kick on the left wing.
Foul by Jonjo Shelvey (Newcastle United).
Ryan Sessegnon (Fulham) wins a free kick in the defensive half.
Foul by Vurnon Anita (Newcastle United).
Neeskens Kebano (Fulham) wins a free kick on the right wing.
Foul by Paul Dummett (Newcastle United).
Offside, Newcastle United. Jesús Gámez tries a through ball, but Matt Ritchie is caught offside.
Goal!  Newcastle United 1, Fulham 3. Daryl Murphy (Newcastle United) left footed shot from outside the box to the bottom left corner. Assisted by Christian Atsu.
Offside, Newcastle United. Christian Atsu tries a through ball, but Matt Ritchie is caught offside.
Substitution, Newcastle United. Yoan Gouffran replaces Dwight Gayle.
Attempt missed. Jesús Gámez (Newcastle United) right footed shot from outside the box misses to the left following a corner.
Corner,  Newcastle United. Conceded by Neeskens Kebano.
Kevin McDonald (Fulham) is shown the yellow card for a bad foul.
Foul by Kevin McDonald (Fulham).
Daryl Murphy (Newcastle United) wins a free kick in the attacking half.
Stefan Johansen (Fulham) wins a free kick in the attacking half.
Teenage full-back Ryan Sessegnon scored twice as play-off chasing Fulham stunned Championship leaders Newcastle with victory at St James' Park.