Results are presented from a semi-coherent search for continuous gravitational waves from the brightest low-mass X-ray binary, Scorpius X-1, using data collected during the first Advanced LIGO observing run (O1). The search combines a frequency domain matched filter (Bessel-weighted $mathcal{F}$-statistic) with a hidden Markov model to track wandering of the neutron star spin frequency. No evidence of gravitational waves is found in the frequency range 60-650 Hz. Frequentist 95% confidence strain upper limits, $h_0^{95%} = 4.0times10^{-25}$, $8.3times10^{-25}$, and $3.0times10^{-25}$ for electromagnetically restricted source orientation, unknown polarization, and circular polarization, respectively, are reported at 106 Hz. They are $leq 10$ times higher than the theoretical torque-balance limit at 106 Hz.
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