The British number one was level at 1-1 in sets with the unseeded Frenchman before rain brought a premature end to play on Thursday.
But play resumed on Friday on Court 5 and Evans forced a tiebreak in the third set to boost his hopes of victory.
However, the 29-year-old followed Cameron Norrie out of the tournament on Court 5 with a 4-6 6-3 7-6 (5) 7-6 (1) loss after nearly hours in total on court.
Evans became increasingly frustrated with distractions from around the stands and on several occasions shouted in direction of the handful of people watching.
But the Briton will be disappointed with his inability to force a deciding set after leading 4-1 in the fourth, only for Moutet to mount a comeback and dominate the tiebreak 7-1.
Norrie also endured a day of frustration against the highly-rated Spaniard, Alejandro Davidovich Fokina after losing in four sets to the 21-year-old.
The 25-year-old lost the first-set tiebreak against Fokina and despite a comeback in the second, saw his level drop significantly as the Spaniard breezed to victory.
The pair were the last British players left in the singles after Andy Murray and Johanna Konta went out in the second round.