The 22-year-old Clutario amassed a six-game total of 1308 to come in third while the fancied Liza Del Rosario submitted a 1304 card to finish fourth.
Koreas Kim Soo Kyun rolling a high 244 in the last game overtook Japans Miyuki Kubotani for the gold with 1362. The Japanese had 1334 in the six-game series for the silver.
"I didnt know what was happening. I just concentrated on my game. Hindi ko inaasahan na mananalo," said Clutario who won the 1997 Asian FIQ masters and the 2001 Malaysian Open titles. She had lines of 198-217-196-234-227-236.
Four-time World Cup winner Paeng Nepomuceno emerged as the best-placed Filipinos in their side of the mens singles but could only finish fourth, his day shattered with a low 178 in the third game. He had a total of 1352. Leonardo Rey tallied 1,238 for 23rd overall.
Clutario was running eighth with two games to go after a stirring 234 on the previous game and chalked up 227-236 in the last two games, overtaking Koreas Kim Hyo Mi for the bronze.
Del Rosario, who won the gold in the recent Asian team championships in Hong Kong, staggered with a 177 in the second game, then produced impressive rounds of 214-213-205-259 for fourth overall.
Four more bowlers were to compete for the gold that had eluded the Philippines the past three Asian Games late Thursday night CJ Suarez, Engelberto Rivero, Chester King and Rowen Jay Bautista in the second phase of the singles event.
"We fell short because we started slow. I think it was more of being the first day and were a bit tense," said national coach Johnson Cheng.
"I think we will see better bowling from our men and women teams now that we have secured the bronze. Our two ladies missed the gold and the silver but they put up some performance and came back very strong. I look to a promising tournament."
Nepomuceno, 42, on the second wind of an illustrious career after undergoing a delicate wrist surgery in the United States, had some problems on lanes 7 and 8 and reeled from a 178 in the third game. He problems on lanes 7 and 8 and reeled a 178 in the third game. He had 226-220-178-203-257-268.
"Malas lang but then this is just the start and theres the masters to look forward to," said Nepomuceno who was satisfied with his 225.33 average in the event.
United Arab Emirates Al Hassan Shaker Ali topped their side of the singles with 1372, follwed by Hongkongs Eric Kwok Yau (1364) and UAE Al Hemeli Hulaiman Mub (1363). Lito A. Tacujan