The former Manchester United starlet unleashed a truly unforgettable 41st minute rocket from 35 yards, curling into the top corner after clipping the underside of the bar and leaving goalkeeper Daniele Padelli no chance.
Twenty minutes into the second half he scored another long-range effort, from just a few yards closer, that flew into the bottom left corner.
Mirko Vucinic then added a third to ensure all three points, and soon after turned provider to set up substitute Alessandro Matri for a fourth as Juventus flew five points clear at the top of the table.
Juventus were without a host of first-choice stars for the clash against eighth-placed Udinese, but any thoughts that the visitors might have an easy ride were forgotten as Juventus stormed out of the blocks to control the match.
Sebastian Giovinco, Emanuele Giaccherini and Paolo de Ceglie were all in irresistible form for the Serie A leaders, and only superb reaction saves from Padelli stopped goals by Arturo Vidal and Giaccherini inside the first 20 minutes.
Just as it seemed that Juventus' domination would prove fruitless in the opening period, Pogba broke the deadlock after picking the ball up in space out on the right. With no defenders closing him down he had a free chance to shoot, and he made the most of it with a strike carved in from long range off the outside of his right boot.
Udinese made a brighter start to the second half and looked like getting back into the match, particularly when Antonio di Natale shrugged off his flu to play the last 40 minutes.
But Pogba stepped up just after the hour mark to put the game beyond Udinese - and once again it was out of nowhere. The 19-year-old took a pot shot from 30 yards out which skidded powerfully in at the bottom left corner to continue his dream night and ensure the crowd cheered for him to shoot even from inside his own half thereafter.
Six minutes later Mirko Vucinic added emphasis to the scoreline as he controlled a corner on his chest, then span and volleyed home despite Pradelli getting a hand to the ball.
Then, with 10 minutes left, Vucinic released Matri with a well-weighted ball which allowed the out-of-favour striker to slot home and complete a magnificent night for Juventus' second-string stars.
Udinese, meanwhile, were not the only losers on the night: Alex Ferguson will be kicking himself at having let a star-in-the making such as Pogba slip through his fingers.