England were clinging to an 11-9 lead courtesy of a first-half George Ford try but struggling to build any cohesive attacks as Italy, with three Carlo Canna penalties in the bag, were holding their own.
Joseph, however, was gifted an interception try after 53 minutes and grabbed his second five minutes later to completely change the game.
He completed his treble with a driving run and Owen Farrell, who also landed three penalties, added an excellent fifth as the game followed the pattern so familiar in the previous 21 encounters, all won by England.
"We thought we’d get our points in the last 20 minutes and that's how it turned out," Jones told ITV.
"We got our rhythm and it was a pretty good performance. We tried to put pace in the game and get them fatigued."
The first half was the usual arm-wrestle that this fixture so often throws up with Ford's sharp try in the corner a rare highlight.
Although Canna landed three penalties, Italy's old failing of poor goalkicking hurt them again as he missed a straightforward shot soon after the restart that would have put them ahead -- and the hosts paid a heavy price.
Under no real pressure, Leonardo Sarto floated a calamitous pass 15 metres in front of his own posts that Joseph picked off for a gift of a try.
Within five minutes he got his second from even closer range, scooping up a grubber by replacement scrumhalf Danny Care.
With the win in the bag England coach Eddie Jones felt able to empty his bench and give debuts to Maro Itoje and Paul Hill, with flanker Jack Clifford winning a second cap.
More forward pressure then got the ball to Joseph but this time he had to use all his strength and determination to force his way through three tacklers and score his third.
Great passing then opened the way for Farrell to score a superb fifth as England, in the second half at least, delivered the "good hiding" Jones had demanded after opening their account with victory in Scotland last week.
England join France on four points from two wins and return to action in two weeks at home to Ireland, while Italy host Scotland in what already looks likely to be the usual wooden spoon decider.
(Writing by Mitch Phillips, editing by: Toby Davis)
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