Old, good news! The Spotify Web Player has been launched quite some time ago. However, the web-based player is yet to graduate from beta and no direct link to it on Spotify official website.
Well, this is the link (http://spoti.fi/18RVIRm) of Spotify Web Player for my own convenience as I always lost the link after clearing the cache of my web browser. The link enables the web browser to access the Spotify Web Player. Hopefully, it will be helpful for you too.
The full featured Spotify Web Player supports Chrome, Firefox, Safari, and Internet Explorer. So you can still enjoy listening songs with Spotify from work or any computer even without Spotify software installed.