Elosegui, a 22-year-old from Spain, leads 16-year-old American Morgan Pressel by one stroke at the 6,365-yard, par-72 course. American Amie Cochran sits two strokes back at 2-under-par 70.
Teenage sensation Michelle Wie struggled through her round, shooting a 3-over 75.
'I'm not even close to satisfied,' said the 14-year-old Wie, the 2003 Women's Amateur Public Links champion. 'It was one of those rounds where you got off to a good start, start making birdies, then I don't know what happened after that. I've been working on my swing and it felt really good in the beginning, but then I lost it.'
2004 U.S. Girls' Amateur winner Julieta Granada fired a 1-over 73, while runner-up Jane Park, 17, joined Wie at 3-over. Park was also the 2003 runner-up in this event.
Last year's U.S. Women's Amateur champion Virada Nirapathpongporn is not competing after turning pro.
The oldest competitor in the event is 70-year-old Marlene Streit. The Canadian, who is a four-time USGA champion, shot 79. She was out-driven on most holes by 40 to 50 yards.
'I think that 36 was like shooting 30,' said Streit of her front nine. 'Once you get near the putting green, it's all the same game.
'I think the only iron I hit was my R-90,' she said, referring to a model of sand wedge popular more than 35 years ago.
Seven-time USGA event winner Carol Semple Thompson, 55, fired an 8-over 80.