The Wales international struck early on in the fixture before putting the result out of Posh's reach with his second just before the break.
New signing from Lazio Valon Behrami made his Hammers bow and looked lively before he was replaced on 66 minutes.
Bellamy's opener came in the third minute when he latched on to Scott Parker's pass before rounding shot-stopper Joe Lewis to tap home.
On 42 minutes, Bellamy, who endured an injury-plagued campaign last term, doubled West Ham's lead via an expertly taken lob over Lewis.
Peterborough came closest late in the second half when George Boyd's effort beat goalkeeper Robert Green but came back off the post.