Virendar Sehwag and Virat Kohli beat a tight field, a spin attack and some bouncers to take India beyond the 300-mark.
Shehwag is also the first centurion of World Cup 2011.
The partnership of over 150 runs started when Sachin and Gambhir departed early.
Sachin had to leave following a crazy mix-up between him and Sehwag and Gautam Gambhir was concentrating on giving the strike to Sehwag when his stumps were ripped apart by spinner Mahmudullah.
Earlier, Bangladesh captain Shakib Al Hasan won the toss and sent India in to bat in the opening match of the World Cup at the Sher-e-Bangla stadium on Saturday.
Speaking right after the toss, Indian skipper Mahendra Singh Dhoni said he would have made the same choice owing to the due factor. India went in with three seamers in Zaheer Khan, Shanthakumaran Sreesanth and Munaf Patel and just one specialist spinner in Harbhajan Singh.
Bangladesh picked two seamers and two frontline spinners, but left out former captain Mohammad Ashraful from the playing eleven.
Bangladesh: Shakib Al Hasan (C), Tamim Iqbal, Imrul Kayes, Junaid Siddique, Raqibul Hasan, Mushfiqur Rahim, Naeem Islam, Mohammad Mahmudullah, Abdur Razzak, Rubel Hossain, Shafiul Islam.
India: Mahendra Singh Dhoni (C), Virender Sehwag, Gautam Gambhir, Sachin Tendulkar, Virat Kohli, Yuvraj Singh, Yusuf Pathan, Harbhajan Singh, Zaheer Khan, Shanthakumaran Sreesanth, Munaf Patel.
Umpires: Steve Davis (AUS) and Kumar Dharmasena (SRI)
Pitch conditions: A hard wicket that is expected to help the slow bowlers in the later stages of the game. The team bowling second could find the ball difficult to grip if late evening dew arrives as predicted.