The 2010 Commonwealth Games gold-medallist defeated Uttar Pradesh's Rahul Gupta to advance to the second round of the welterweight (69kg) category.
Mandeep, on the other hand, got the better of Tamil Nadu's RP Prakash Raj.
The 2014 Commonwealth Games silver-medallist is returning to action after recovering from a wrist injury and made his debut in the middleweight 75kg category.
Also advancing was King's Cup bronze-medallist Rohit Tokas, who defeated Tripura's Vishnu in his light welterweight (64kg) opening bout.
Over 300 boxers from 37 states and sports promotion boards are competing in the championships.