no the 6300 is not more powerful than an 8350 in any regard at stock clocks, and typically can't go any higher than an 8350 when overclocking. on a single core basis, clock for clock, they are the same.
I'd recommend the 8350 if you can afford it. It has one more module (which gets you one more FPU which is pretty important for gaming) and games are starting to utilize 8 cores now so yeah, the 8350 will win in newer games, and in games where having more than 3 FPUs available will be beneficial (all quad-core optimized games and up).
I don't recall seeing any benchmarks that showed the 6300 beating an 8350 (this even includes dual core optimized games). At best the 6300 was a bit behind the 8350, and more often than not, the gap is substantial enough.
Don't get me wrong, the 6300 is a great cpu, and has tons of performance for its lower price. But in the scope of your question the 8350 wins hands down. If you are looking to save a bit on the cpu i'd recommend the 8320. The performance difference would be much less, and if you overclock to 4.0 then you'll be getting the same performance as an 8350