Bjarne's book is fantastic, especially for C++ syntax, but the one book that will really make you a competent C++ programmer is Meyers' Effective C++. Get it. Read it.
I as well do not have a CS degree, but I work for a silicon valley startup. It is possible, you just have to be aware of what's out there and never stop learning. Many students who graduate with a computer science degree end up working in a language they didn't study, so be sure to hit the fundamentals. If you hear something that's unfamiliar to you, be sure to find a good book and a coffee shop and get to it. The C++ will come in time - with Stroustrup and Meyers, you've got 90% of what it takes to be good at C++
