We consider the \textsf{Unit Interval Selection} problem in the one-pass random order streaming model. Here, an algorithm is presented a sequence of $n$ unit-length intervals on the line that arrive in uniform random order, and the objective is to output a largest set of disjoint intervals using space linear in the size of an optimal solution. Previous work only considered adversarially ordered streams and established that, in this space constraint, a $(2/3)$-approximation can be achieved, and this is also best possible, i.e. any improvement requires space $Ω(n)$ [Emek et al., TALG'16]. In this work, we show that an improved expected approximation factor can be achieved if the input stream is in uniform random order, with the expectation taken over the stream order. Specifically, we give a one-pass streaming algorithm with expected approximation factor $0.7401$ using space $O(|OPT|)$, where $OPT$ denotes an optimal solution. We also show that algorithms with expected approximation factor above $8/9$ require space $Ω(n)$, and algorithms that compute a better than $2/3$-approximation with probability above $2/3$ also require $Ω(n)$ space. On a technical note, we design an algorithm for the restricted domain $[0,Δ)$, for some constant $Δ$, and use standard techniques to obtain an algorithm for unrestricted domains. For the restricted domain $[0,Δ)$, we run $O(Δ)$ recursive instances of our algorithm, with each instance targeting the situation where a specific interval from $OPT$ arrives first. We establish the interesting property that our algorithm performs worst when the input stream is precisely a set of independent intervals. We then analyse the algorithm on these instances. Our lower bound is proved via communication complexity arguments, similar in spirit to the robust communication lower bounds by [Chakrabarti et al., Theory Comput. 2016].