That's what I was suggesting, certainly.
The only thing I would add in your particular case is that the airline check-in counters may find it a little strange if you present them with your US passport - they're checking to see whether you've got a valid visa for the country you're entering, because if you haven't, then they may possibly be fined for carrying you there.
(This is of course subject to the immigration laws of the country you're travelling to. Several countries allow visa-free entry to certain countries' citizens for a certain amount of time; others have facilities to issue visas on arrival; others insist that you have a valid visa before you arrive.)
In your case (and in any case where you need a valid visa to enter your destination country) you're probably better advised to show the check-in staff the passport you're planning to use at the other end of your flight. But you should still show the passport control officials the passport you used when you entered the country you're in the process of leaving (ie their country) - since they're interested in knowing whether you're in their country legally.
So for your circumstances, you show the check-in staff the UK passport with the Indian visa; you then go through passport control and show your US passport (the one you used to enter the US with).
I'm not an immigration official, so the usual disclaimers apply. But that is my understanding of the procedure, at least.