No, you don't need to open source your application.
You only need to open source a work derived from a GPL work when you distribute the work. But when your program runs your own server, you are not distributing it to your users. That means you do not need to license it to them. You can even modify the sourcecode. As long as the software doesn't leave your server, the sourcecode doesn't have to leave it either.
There is the AGPL which closes this loophole by requiring to sublicense the work to everyone who interacts with the software via a computer network, but the normal GPLv2 does not include this clause.