One sided confidence intervals are dual to one tailed hypothesis tests just as regular two sided CIs are dual to two tailed tests.  

If $\theta$ is a parameter, and we say that $(a,\infty)$ is a one sided CI for $\theta$, then this means that $a$ was found by a process that will yield a value below the true value of $\theta$ $95\%$ of the time. 

In your case, the parameter of interest is the difference of means: $\mu_x-\mu_y$.  If you construct a one sided confidence interval for this parameter, of the form $(a,\infty)$, then you can say with 95% confidence that $a<\mu_x-\mu_y$.  This, if $0\leq a$, you may reject the null hypothesis.