Opener Sikandar Raza top scored for Zimbabwe with 82 and held up the innings from one end.
Making his ODI debut for India, Jaydev Unadkat dismissed Brendan Taylor for 12 to claim his first ODI wicket.
Amit Mishra was India's most successful bowler picking up three wickets including that of Raza.
Virat Kohli steered his team to victory with his 15th ODI century scoring 115 off just 108 balls to give India 1-0 lead in the five-match series. (AFP Images)
Virat Kohli led from the front with 115. This was Kohli's 15th ODI century
Ambati Rayadu became the 11th Indian to score a half-century on ODI debut. His 63 not out ensured India reached the target easily.
Both Kohli and Rayadu were involved in a 159-run partnership for the fourth wicket after India were two down for 57.