It turns out that someone else on StackExchange asked about t-tests and sample sizes, and the summary appears to be that yes, the t-test is valid even in small sample sizes. You merely need to approximately satisfy the t-test's assumptions. So, you could just go ahead and do a t-test. Your power is what it is with a sample of 10 in each arm.
I've heard one professor say that permutation tests are appropriate for inference in small samples. I haven't yet found anyone saying that they are better than t-tests in small samples, but they are basically only practical in smaller samples, and the link below says they can be a good check of a t-test result. In a permutation test, you would (by my reading of the link) assume that the between-group difference is actually zero, then using your data, calculate all possible permutations of the between-group difference. You're basically calculating an exact p-value for the difference - remember that p-values mean that assuming the null hypothesis is true (i.e. that the difference between groups is zero), what's the probability that you'd have seen the between-group difference you got in your sample?
