The storm had already claimed the lives of at least two people in China's Hainan and 16 people in the Philippines
Westwood, who turns 47 in April, shot a seven-under par 65 to move one shot ahead of Austria's Bernd Wiesberger, who also shot a 65, and overnight leader Francesco Laporta of Italy.
Matthew Fitzpatrick birdied his last two holes in a round of three-under par 69 to go fourth at 12-under par 204, while American Kurt Kitayama (68) and Spain's Sergio Garcia (69) were tied fifth one shot further behind.
World number one Brooks Koepka (70), coming back to competitive golf after a 14-week layoff due to stem cell treatment for his left knee injury, was in a tie for 48th place at five-under par 211.
England's Westwood produced his eagle on the par-five eighth hole, smashing a hybrid iron from 266 yards to less than three feet.
His round also contained six birdies and a solitary bogey. The dropped shot on the 14th hole, where he had to stand in a bunker with the ball outside in the rough, was his only bogey in the last 37 holes.
"I played well. I holed a couple of putts early on which gave me some momentum and then hit a brilliant shot into the eighth. Then backed it up with a birdie at the ninth, so a bit of momentum, got on a roll," said Westwood, winner of 24 titles on the European Tour.
Westwood was winless last year but had finished 2018 in style when he won the NedBank Challenge in South Africa.
He is currently ranked 63rd in the official world golf ranking, and even though he has qualified for the Masters this year, getting back into the top-50 would get him into all the majors and WGC events.
Even though a lot rides on a good performance early in the season, Westwood said he rarely played golf or hit balls in the off season.
"Coming here, I expected to have to work hard on my game and scramble a lot. I don't play a lot when I'm away from tournaments now," he said.
After making a birdie on the 13th, the 34-year-old Wiesberger burned the edges on the next two holes with his birdie attempts, but managed to close with three in a row from the 16th onwards.
"I struggled the first two days. I think I hit more fairways today and I can then get myself into positions to take on the flags, which I did well," said the world number 23.
Laporta made an early bogey on the par-five second hole and was one-over through seven before making four birdies in the last 11 holes.
"I was a bit nervous on the first couple of holes and made a bad mistake on second when I pulled my second shot," said the reigning Challenge Tour Order of Merit champion.
He added: "Tomorrow, it would be just a dream. I just want to think about my score and my game. I am trying not to see the leaderboard. I think it's going to be a good day tomorrow."Scores
202 - Lee Westwood (ENG) 69-68-65
203 - Bernd Wiesberger (AUT) 69-69-65, Francesco Laporta (ITA) 71-63-69
204 - Matthew Fitzpatrick (ENG) 68-67-69
205 - Kurt Kitayama (USA) 67-70-68, Sergio Garcia (ESP) 67-69-69,
206 - Thomas Detry (BEL) 69-70-67, Rafael Cabrera-Bello (ESP) 67-68-71
207 - Sebastian Heisele (GER) 70-73-64, Jack Senior (ENG) 71-70-66, Ross Fisher (ENG) 70-71-66, Zander Lombard (RSA) 67-71-69, Louis Oosthuizen (RSA) 69-68-70
208 - Kiradech Aphibarnrat (THA) 70-73-65, Tommy Fleetwood (ENG) 71-70-67, Mike Lorenzo-Vera (FRA) 68-73-67, Romain Langasque (FRA) 70-71-67, Joakim Lagergren (SWE) 75-65-68, Victor Perez (FRA) 70-70-68, Jack Singh Brar (ENG) 70-70-68, Andy Sullivan (ENG) 60-71-68, Scott Hend (AUS) 69-70-69, Thomas Pieters (BEL) 68-70-70
209 - Brandon Stone (RSA) 71-72-66, Paul Waring (ENG) 71-71-67, Sam Horsfield (ENG) 73-69-67, Jordan Smith (ENG) 74-68-67, Ashley Chesters (ENG) 72-69-68, Edoardo Molinari (ITA) 68-73-68, Shaun Norris (RSA) 64-77-68, Branden Grace (RSA) 68-72-69, Matthias Schwab (AUT) 74-66-69, Richie Ramsay (SCO) 71-69-69, Yuxin Lin (Am-PRC) 69-70-70, Zach Murray (AUS) 67-72-70, Patrick Cantlay (USA) 71-66-72
210 - Gavin Green (MAL) 73-70-67, Scott Jamieson (SCO) 71-72-67, Jorge Campillo (ESP) 71-72-67, Sean Crocker (USA) 72-70-68, David Lipsky (USA) 72-69-68, Martin Kaymer (GER) 68-73-69, Erik Van Rooyen (RSA) 73-68-69, Shubhankar Sharma (IND) 71-70-69, Soren Kjeldsen (DEN) 70-70-70, Masahiro Kawamura (JPN) 69-71-70, Li Haotong (PRC) 67-69-74
211 - Brooks Koepka (USA) 66-75-70, Joost Luiten (NED) 70-68-73
The storm had already claimed the lives of at least two people in China's Hainan and 16 people in the Philippines
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