It sounds like that is what is occurring on the system. One way to possibly avoid this would be to uncheck the option for "Release ASIO driver in background" within Cubase. However, then you would not be able to hear audio from the web page.
The best solution is to use an audio interface that has a dedicated ASIO driver. This will result typically in better latency, performance, reliability, and sound quality. There are many ASIO devices from Steinberg, Yamaha, Tascam, Presonus, Alesis, RME, MOTU, Lexicon, TC Electronic, MAudio, and the list goes on and on.
Personally I would recommend our UR22 for an inexpensive but great sounding device. http://www.steinberg.net/en/products/au ... start.html
But I'd also recommend going to a local music store's sound recording department and finding the device that fits your needs and budget. I hope this information helps.