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."

The 68-year-old from Bournemouth died when his Vauxhall Astra hit a Nissan Juke on the A31 near Corfe Mullen.
The road, near the Coventry Arms pub, was closed for more than five hours after the crash happened at 13:20 BST.
The driver of the Nissan, a 49-year-old woman from Waterlooville, was seriously injured and airlifted to Southampton General Hospital but police said her condition was not life-threatening.
Three men who were travelling in the Nissan sustained serious but not life-threatening injuries and were taken to Poole Hospital.
Sgt Mark Scammell said: "I would like to thank motorists for their patience during the road closures which are necessary to allow emergency services to deal with the incident."

The deal will see the firm taken over by holding company Cheil Industries, another part of the Samsung group.
The merger is strongly opposed by some of Samsung C&T's shareholders, led by US hedge fund Elliott Associates.
For Samsung's founding family, the move is a crucial step in consolidating control of the conglomerate.
Shareholders in Cheil Industries approved the merger earlier on Friday.
Shares in Samsung C&T fell 10.4% and Cheil Industries dropped 7.7% after the merger was approved.
Elliott Associates, which is the second largest single shareholder in C&T, says the takeover significantly undervalues the company's stock.
The hedge fund had filed several unsuccessful law suits to stop the vote from going ahead.
The takeover is key to consolidating the Samsung founding family's control of the multi-headed conglomerate.
It is of particular significance as it comes ahead of a generational power transfer at Samsung.
The business empire's patriarch Lee Kun-hee has been in hospital since May 2014 and his son Lee Jae-yong is to take a bigger leadership role.
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.
[NEXT_CONCEPT]
A man has died in a crash involving two cars on one of Dorset's busiest routes.
[NEXT_CONCEPT]
One of South Korea's most controversial mergers has been given approval by shareholders of the construction company Samsung C&T.