There were no breaks of serve during the opening set, which Monfils claimed 7-5 in a tie-break.
The Frenchman was the first to blink in the following set, as Lu broke in game nine before holding to love to level the match.
Monfils claimed only 12 points during set three, which he lost without winning a single game.
However, the tables were turned in the fourth set, as Monfils stormed to a 6-1 success in just 26 minutes.
The first game of the final set lasted 17 minutes before Lu held serve, but Monfils quickly got over his frustration by breaking to 40 at the next attempt.
Moments later, Lu broke for a fifth time to level the match and the Taiwanese then began to serve with much more authority.
He twice forced Monfils to hold to stay in the match, which the Frenchman managed in less-than-convincing fashion.
Lu's serving frailties returned in game 13, as Monfils edged back in front with a break to 30.
Monfils converted his fourth match point and held to 40 in the following game to seal a 7-6 4-6 0-6 6-1 8-6 win in a contest that lasted over three-and-a-half hours.
Fellow Frenchman Gilles Simon will provide the opposition for Monfils in the third round.