Fearnley has enjoyed a meteoric rise in the last 12 months, rising from outside the top 500 to now sitting at 60th in the world.
This was just his sixth tour-level match on grass but it did not look like it as he played some brilliant tennis at times that Moutet had no answer for.
The British number two came back from 0-40 down in his first service game of the match before immediately breaking Moutet to take control of the opener.
It looked like a straightforward afternoon for Fearnley as Moutet grew increasingly irked but the Frenchman regained his composure to dominate the second set, the momentum swinging after three double faults by the Scot in his first service game.
However, Fearnley found an extra level in an excellent third set and Moutet had no answer to his big hitting, with his frustrations boiling over as he received a time violation after arguing with the umpire.
But Fearnley kept his composure well to serve out victory and join compatriot and British number one Jack Draper in the last eight.
