The result has left Huddersfield in 16th position in the Premier League table - four points ahead of 18th-placed Southampton, who visit Arsenal on Sunday.
Moments later, Albion had another opportunity when Propper found Solly March in a dangerous position, but the 23-year-old saw his shot blocked.
The pressure was intense from the home side in the first period, and they took the lead in the 29th minute when March's shot from the edge of the box hit the post before bouncing into the back of the net off Huddersfield goalkeeper Lossl.
Brighton came close to regaining the lead early in the second half when Lewis Dunk met a deep free kick from March, but the centre-back's effort just dropped wide of the post.
Huddersfield were a threat though, and Mounie headed over the crossbar just past the hour before Tom Ince tested Ryan from distance.
The match swung in the away side's favour in the 74th minute when Propper picked up a straight red card for a late challenge on Jonathan Hogg.
Brighton were still creating chances, however, and Lossl did well to deny Schelotto 10 minutes from time, before Duffy headed the crossbar from close range moments later.
Mounie and Ince then took it in turns to have half-chances for Huddersfield as the match headed for its final moments at 1-1. The visitors looked the team more likely to win it in stoppage time, but the points were ultimately shared on the afternoon.
BRIGHTON (4-1-4-1): Ryan; Schelotto, Duffy, Dunk, Bong; Propper; Izquierdo, Stephens, Gross (Kayal 76'), March; Murray (Ulloa 67')
HUDDERSFIELD (4-4-1-1): Lossl; Kongolo, Schindler, Zanka, Hadergjonaj; Van La Parra (Quaner 90'), Mooy, Hogg (Billing 80'), Ince (Depoitre 90'); Pritchard; Mounie