The visitors closed on 293-9 at the end of day three, with a lead of 260.
Jacques Rudolph's wait for a first Championship hundred in 2016 continued as he chopped onto his stumps for 56.
Earlier, Essex had been bowled out for 319, with Craig Meschede (5-84) taking his first five-wicket haul in the Championship this season.
The home side secured promotion and the second-tier title by passing 250 in their first innings, thanks largely to a century from captain Ryan ten Doeschate, who added eight to his overnight score before falling for 117.
With the ball, Ten Doeschate (2-20) removed both Rudolph and Bragg (54) in his first over, the latter having passed 1,000 Championship runs for the season, after the pair had shared a second-wicket stand of 89.
Glamorgan lost six wickets for 36 runs but Mark Wallace (75 not out) and Timm van der Gugten (32) helped push their advantage above 250.
Glamorgan batsman Will Bragg told BBC Wales Sport:
"The four-day game can ebb and flow, but Wally (Mark Wallace) showed some great resilience and put us in a competitive position going into the last day.
"I find it's hard to be consistent in a long season so it's pleasing from that point of view (to reach 1,000), I find more consistency batting at three.
"I've felt like I haven't given my wicket away much at all this year which is pleasing.
"From a team point of view it's been disappointing but there are positive signs with the youngsters coming through and we've a lot to build upon."
Division Two champions Essex will need to chase a sizeable target on the final day to beat Glamorgan at Chelmsford.