England's highest-scoring win in a Calcutta Cup fixture, which at the same time prolonged Scotland's misery in the English capital past the 34-year mark, began in style as Joseph ran through with just three minutes on the clock.
Owen Farrell converted and then added a penalty soon after for the first of his 26 points in the one-sided affair, before Joseph slalomed through for a second with a little over a quarter of the match played.
There was to be some respite for the visitors when Gordon Reid found the whitewash, but another Farrell penalty - this one from 44 metres out - and a Watson try put England well on their way to a famous victory.
Within two minutes of the restart, Joseph again darted through for his hat-trick, with Vunipola - back in an England shirt following a three-month layoff - crossing over on the hour to reach the milestone figure.
Huw Jones managed to find a way past the England backline twice in the space of 20 minutes to give Scotland's fans a little something to cheer about, but Care weaved through for two of his own late on to put the icing on the cake.
England now head to Dublin next weekend with a Grand Slam in their sights, as well as overtaking New Zealand's outright record of 18-successive victories.