Well there's your mistake. If you apply through their website, you're instantly being funneled into an HR tool that weeds out anyone without a BS in CS.

The trick is to get in contact with that company's recruiters using a site like LinkedIn. Establish a bit of rapport, send your resume and portfolio, and let them handle setting up your phone screening and technical on-site.

Recruiters are an amazing resource in our industry and are great for getting your foot in the door... Especially when you're being blocked by those automated HR filters.