We investigate the occurrence of powers of tight Hamilton cycles in random hypergraphs. For every $r\ge 3$ and $k\ge 1$, we show that there exists a constant $C > 0$ such that if $p=p(n) \ge Cn^{-1/\binom{k+r-2}{r-1}}$ then asymptotically almost surely the random hypergraph $H^{(r)}(n,p)$ contains the $k$th power of a tight Hamilton cycle. This improves on a result of Parczyk and Person, who proved the same result under the assumption $p=ω\left(n^{-1/\binom{k+r-2}{r-1}}\right)$ using a second moment argument.