But Coppejans, ranked 278th, fought back and broke Murray early in the second set, only for the Scot to rally and level things up at 3-3.
The rest of the second set stayed on serve, but it was two-time Wimbledon champion Murray who held his nerve to win a tense tie-break 7-4 and take the match.
'One of the most important players on the tour' - Nadal hoping Murray comes back
Up next for Murray is either eighth seed Pablo Cuevas or Bolivia's Hugo Dellien, who meet in the first round on Wednesday.
Murray, who won his first ATP Tour matches since his surgery during the Asian swing and reached the quarter-finals of the China Open, is hoping to regain full match fitness before he makes his much-anticipated Grand Slam return at next year's Australian Open.
South Korea's Kwon Soon-woo earlier rallied past Frenchman Richard Gasquet 1-6 6-3 6-4 while Feliciano Lopez beat Cameron Norrie 7-6(2) 6-4.
Also advancing were Germany's Jan-Lennard Struff, France's Gilles Simon and Italy's Jannik Sinner.