Pegged as relegation favourites after ensuring promotion via the Championship play-offs, David Wagner's men dictated matters in the first half and made top-flight history as they broke the deadlock in the 23rd minute.
Joel Ward's own goal saw Huddersfield become the first team to net their maiden Premier League goal in that manner.
The lead was doubled just three minutes later through Mounie's fine header and the hosts were lucky not to see De Boer's first game in charge take a further turn for the worse before half-time.
Palace did improve after the second-half restart in a bid to avoid an opening-day defeat for De Boer, who was sacked after only 85 days in charge in his previous managerial stint at Inter Milan.
The efforts of Christian Beneteke and Wilfried Zaha were not good enough to haul Palace back into the game, and Mounie's assured first-time finish made sure of the points 12 minutes from time.
Tom Ince, the only player in the Huddersfield starting XI with Premier League experience, saw a right-wing cross fall to him inside two minutes, but could only direct his half-volley straight at Wayne Hennessey.
Huddersfield continued to have the better of the proceedings and, after Hennessey had comfortably kept out Philip Billing's long-range drive, the visitors found the net in the Premier League for the first time.
Cristopher Schindler flicked Mooy's corner to the far post, where Mathias Jorgensen's off-target effort found the net via the unfortunate Ward.
And Mooy - making his Premier League debut - was the creator again as Huddersfield doubled their lead in much more aesthetically pleasing fashion.
The Australia midfielder, who played a full game for the visiting side, produced an excellent left-wing delivery and Mounie rose brilliantly to power beyond a helpless Hennessey and into the top-left corner.
Jorgensen should have made it 3-0 when he was left unmarked at a corner, but the Denmark defender could only direct his header well wide of the target.
Zaha then spurned Palace's first real sight of goal as he pounced on a defensive mistake, only to see his low shot turned wide by Jonas Lossl and not receive a corner for his efforts and Benteke headed wide from a corner just before half-time, with his subsequent frustration a fitting encapsulation of the mood at Selhurst Park at the interval.
Andros Townsend replaced the injured Luka Milivojevic at the start of the second half for Palace, and the hosts did the pressing early in the second half, Benteke superbly denied by a diving save from Lossl after a Patrick van Aanholt corner.
Lossl produced a further demonstration of his reflexes to keep out a Zaha effort at the near post but only a fine recovery by Timothy Fosu-Mensah prevented Mounie from firing in again at the other end after Scott Dann had lost possession.
Dann blazed over in Palace's last chance to halve the deficit, with Mounie giving Huddersfield further reason for jubilation as he beat Hennessey's glove from Collin Quaner's pull back.
Key Opta stats:
- Crystal Palace have won only one of their last six opening-day matches in the Premier League, losing four of the last five (W1 D1 L4).
- Huddersfield are the first team to win on their Premier League debut since Blackpool in August 2010 (4-0 v Wigan Athletic).
- The Terriers are just the fifth newly-promoted side to win on the opening day of the Premier League since 2008-2009 (also West Ham United in 2012, Blackpool in 2010 and Hull City in 2008 and 2016).
- Frank de Boer has enjoyed just one win in his last seven league games as a manager (W1 D1 L5 with Inter Milan).
- Only one of the eight Dutchmen to manage in the Premier League has won in their first game in the competition (Guus Hiddink).
- Huddersfield are the first side to see their first-ever Premier League goal come via an own goal.