His career-best lifted the Bears to 435 despite Tim Bresnan bagging his first five-wicket haul in three years.
Bresnan's 5-85 and James Middlebrook's 3-96 kept Yorkshire in the match, but spinner Jeetan Patel then removed openers Alex Lees and Will Rhodes.
Yorkshire dipped to 88-3 before partially recovering to 128-3 at stumps
The duel between last season's champions and runners-up is going the way of the latter so far with only Westwood, whose 196 came from 377 balls, with 25 fours, having passed 50.
The left-hander's marathon effort finally ended when he fell lbw to off-spinner Middlebrook to provide the 37-year-old with his first wicket since rejoining Yorkshire as emergency spin-bowling cover - and his 50th in first-class cricket for his native county.
Badly depleted by England calls and injuries, Yorkshire went into bat requiring 286 to avoid the follow on and reached 41 without loss before Patel trapped Alex Lees lbw for 19 in the last over before tea.
Warwickshire then struck twice in seven balls as Will Rhodes (46) chipped Patel to mid-on.
Boyd Rankin took the catch and then added a wicket with his second ball, but it was a controversial one as Cheteshwar Pujara was adjudged caught by wicketkeeper Peter McKay who failed to take the ball cleanly, before scooping it up at the second attempt.
After conferring, the umpires sent Pujara on his way to loud boos from a section of the crowd.
Yorkshire batsman Will Rhodes:
"I am really pleased how things have gone in my past three innings and in my opinion I think I should have had a hundred by now, although I did give it away today. But I like to be aggressive against the spinners.
"I am not taking anything for granted. All I can do is put my name in the hat and hope to get selected. If not, I will be happy to go back in the second team for a while."
Warwickshire director of cricket Dougie Brown:
"We have to come back tomorrow and get stuck in again. The third day is always the 'moving' day -  bit like in the Masters - and we have got two days left to try to force a win."
(On Pujara's dismissal) "Peter McKay was convinced that he had caught the ball, even though it may have looked as if he hadn't. Peter is a very trustworthy guy and you have to go with him."
Yorkshire came under pressure at Headingley after Warwickshire took early wickets to exploit the pressure built by Ian Westwood's 196.