Peter Reid's men had been well-beaten 3-0 by Atletico de Kolkata in their opening fixture, but they responded in style as Moritz played a starring role on Saturday.
The former Crystal Palace and Bolton Wanderers man scored twice in the opening 27 minutes as the home side took control, and Mumbai were three to the good at half-time thanks to Subhash Singh's effort.
Moritz completed his hat-trick with 19 minutes remaining, before Johan Letzelter rounded off a comprehensive victory for the home side.
Atletico de Kolkata remain at the top of the early table despite losing their 100 per cent record in a 1-1 draw at home to Delhi Dynamos.
Antonio Lopez Habas' side had defeated NorthEast United 2-0 on Thursday thanks to goals from Fikru Tefera and Jakub Podany, but they were unable to make it three wins from three as Delhi departed with a share of the spoils.
The home side had gone in front through Jofre's 49th-minute penalty, but were pegged back 16 minutes from time as Pavel Elias struck to earn a point for his side.
The game ended on a sour note for Kolkata who, after seeing Borja sent off in their previous outing, had substitute Rakesh Masih dismissed in the closing stages after picking up two cautions in six minutes.
Elsewhere, NorthEast United needed Koke's controversial penalty to salvage a 1-1 draw at home to Goa.
After Gregory Arnolin had opened the scoring for Goa, Debabrata Roy was adjudged to have fouled Durga Boro inside the area, and Koke sent Jan Seda the wrong way from the spot.
