I have purchased a new Laptop. I was away for my work, then my cousin opened the laptop and charged for just some time. The battery was not charged completely.
This resulted that my Laptop couldn't last for longer time because of low battery even it was full later. How can i make my battery work longer?

Run the battery all the way down and then charge it overnight, I doubt that much damage was done to it. Otherwise you just need to get a new battery.

Unfortunately some batteries have memory so if you continually recharge the battery after it has only been 50% discharged, the battery will soon think when it's been charged to 50%, it's full. The solution as suggested by Redsky is to run the battery all the way down before charging.

As you have bought a new laptop you can complain it to the customer care
and i think that you will get a free service,so don't worry

As mentioned, you could try to run the battery all the way till it goes flat, than recharge it again. If that still doesn't work, as descorpio1 said it is still a new laptop so it should be there is a possibility that the battery could be replaced for free.