Deepak out-punched Malaysia's Muhammad Abdul Qaiyum Bin Ariffin 5-0 to advance to the round of 16 in the men's 51kg event.
Nishant notched up a win over Nepal's Dipesh Lama by a similar margin in the men's 71kg opening round.
However, Arundhati Choudhary bowed out in the opening round of the women's 66kg event. The young Indian pugilist was no match to world champion Yang Liu of China, who jabbed and hooked her way to a dominant 5-0 win.
Deepak, who had pipped last year's gold medallist Amit Panghal to make the squad, was involved in an energy-sapping bout.
He used his powerful punches, lunging forward to jab his opponent who was equally fast. Deepak wore down the Malaysian with body shots while gliding across the ring to defend.
Deepak will face the 2021 world champion Japan's Tomoya Tsuboi in the next round.
Competing for the first time since his World Championship bronze medal in May, Nishant used his long reach to land decisive blows on his opponent. The Nepal boxer did well but Nishant's powerful blows forced the referee to give Lama the eight count twice.