Trott hit 175 as Warwickshire totalled 413 at Taunton, backed by half-centuries from Tim Ambrose (63) and Rikki Clarke (57).
Young paceman Grant Thornton then took two of the four Somerset wickets to fall on his Championship debut.
But Marcus Trescothick was still there at the close on 41, out of 94-4.
After two badly rain-interrupted days. the draw remains the most likely result.
But the Bears, so badly beaten by an innings in their opening two matches, have at least given themselves a chance of forcing victory, although it would take 16 wickets in the day to do it - and getting Trescothick out twice.
Bottom club Somerset's target on day four will simply be to avoid the follow-on - and they need a further 170 to do that.
Somerset's Marcus Trescothick told BBC Radio Bristol:
"One of the office staff told me around 10 days ago how close I was to 25,000 runs and I have been sweating on it since then.
"I knew when I went in that I was 29 away. It got a bit tense as I moved closer.
"It is an achievement I am proud of but there is still a big job for me to do in this game.
"It wasn't a great day for us and we need to concentrate hard tomorrow to make sure we come away with a draw."
Warwickshire's Jonathan Trott told BBC Coventry & Warwickshire:
"It's great that Marcus is still playing county cricket and setting an example to the young players in his dressing room and opposition teams.
"There is no chance I will be still be playing at 41. That's just silly! His will be a huge wicket for us tomorrow.
"I had just got past 100 when the new ball was taken so it was important I hung around for longer if we were to get a decent score. I started to think in terms of 400 when Rikki Clarke came in and played so well.
"There is a bit of turn because it is a used pitch. With so much rain, the crustiness has taken longer to form, but hopefully Jeetan Patel can cause them a few problems."
Warwickshire took command of the battle of Division One's bottom two after Jonathan Trott had reached the 42nd first-class century of his career.