Yes you can travel with the old passport having the valid Indian visa along with your new passport and have to present both. My last three trips to India were done this way however I'm thinking of transfering the visa to my new passport because my old passport is the 48 page variety with additional sets of extension pages (24 pages each extension) added and it is bulky. Some of the immigration offices spend additional time scrutinizing the various entry stamps on the old fat passport.
To transfer a visa you use the same form as the one used to get a new visa however the transfer fee is small. You don't indicate where you live but
here is the application from the SFO area consulate office. I think you'll need to use the form from the office that has the jurisdiction for the area you reside in or send the application to.
You'll note in the application form that it tells you that you can travel with the valid visa on the old form along with your new passport.