Defender Elphick gave the Seagulls a 26th-minute lead by leaping to guide a corner from Elliott Bennett in via the far post.
Dicker made the points safe in magnificent style by thundering in a 25-yard volley five minutes after the break.
Exeter improved after a poor first half but carved out few clear-cut chances.
The Grecians threatened for the first time just after the hour when midfielder Liam Sercombe shot just wide before substitute Richard Logan put a good chance into the side netting.
Midfielder Neil Saunders should have done better for Exeter late on when putting a free header wide.