Replacement Mogg claimed the only try of the first half, crossing the line in the corner after a neat passing move by the French team who secured their first victory in a major European competition.
Three Nick Evans penalties, however, kept Harlequins in the game against a side coached by South African Jake White and the English side trailed 13-9 at halftime in director of rugby Conor O'Shea's last game in charge before taking over as Italy coach.
Flyhalf Demetri Catrakilis kicked eight points in the opening period and, after Mogg had caught a high kick to the right corner to grab his second try, the South African converted to stretch Montpellier's lead to 20-9.
He notched two more penalties for Montpellier, who are second in the French Top 14 standings, while Marland Yarde's late try was a mere consolation for Harlequins.
Defeat completed a depressing week for the London club who were rocked by the death of their young prop Seb Adeniran-Olule in a car crash on Monday.
(Reporting by Ed Osmond, editing by Ian Chadband)
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