After Dwight Gayle had given the visitors a surprise lead, Pogba produced a goal and assist to help the Red Devils turn things around by the 54th minute.
Romelu Lukaku then got on the scoresheet to end his seven-game barren run at club level as United romped to a victory which takes them back to second in the table.
Lukaku scored three times for Belgium over the international break but it appeared as though his United drought may continue when he missed the second of two good first-half chances by heading over from Lindelof's cross.
United's delivery into the box was outstanding all game, and it was Pogba's chip onto the head of Anthony Martial which helped them draw level on 37 minutes, with the Frenchman leaping above Yedlin and finding the bottom corner.
Newcastle were unlucky to be behind at half time and could have been level had Isaac Hayden not shot straight at De Gea after working a good chance for himself inside the penalty area.
Pogba made the Magpies midfielder pay by tapping home from close range after being teed up by Marcus Rashford's delicate cushioned header shortly after the break.
United were electric going forward in the second half and the icing on the cake for Jose Mourinho was a much-needed goal for Lukaku on 70 minutes.
Shelvey came close to pulling one back for Newcastle when his low shot nearly caught out De Gea as it skimmed the post.
Zlatan Ibrahimovic joined Pogba in making his comeback, after seven months on the sidelines, and would have marked it with a spectacular goal on the volley had it not been for Elliot, who also saved from Marouane Fellaini late on.
Mourinho's side move back to within eight points of leaders Manchester City, who beat Leicester City 2-0 this afternoon.