Ashley Barnes made a bright start to the match, creating two chances inside the opening seven minutes, but neither of his finishes was on target.
The home side began to find their way into the contest as the first half progressed, but Sam Hoskins saw his strike saved before Byron Webster's header from point-blank range went wide.
Hoskins tested Tomasz Kuszczak again at the start of the second half, but the Brighton keeper continued to frustrate the striker and tipped an Edward Upson attempt wide just five minutes later.
Leroy Lita was brought on after Barnes picked up a knock midway through the second half and his fresh legs made an immediate impact, stretching the home side's defence, but he could not create the opening goal.
Yeovil looked the more likely of the two teams to snatch a late goal, as Shane Duffy headed over just four minutes from time, before Andy Williams and Hoskins both missed the target in stoppage time.
Despite taking a point, Yeovil slip to the bottom of the Championship table.