Parsis (Parsees) fled Persia, in the 11th century (religious persecution, apparently), and finally settled in the Bombay region of India. Dhansak and Patia are typical Parsi (Parsee) dishes.
Lee & Perrins made their Worcester sauce, for the Governor of Bengal at the time, around 1835.
It's quite easy to see why the sauce might be used in a dhansak, given it's ingredients (sweet and sour)