Results are presented from a semicoherent 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. The search combines a frequency domain matched filter (Bessel-weighted 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\%) = 4.0 x 10(-25), 8.3 x 10(-25), and 3.0 x 10(-25) for electromagnetically restricted source orientation, unknown polarization, and circular polarization, respectively, are reported at 106 Hz. They are <= 10 times higher than the theoretical torque-balance limit at 106 Hz.