According to then-contemporary news reports, the first women’s soccer match at Old Wembley Stadium, England’s most-hallowed soccer ground, was contested on July 16th, 1987, a full sixty-four years after the Stadium opened in 1923. In that match, which was the opening game of the fourth annual London International Football Festival, the Cincinnati Cardinals, an under-14 girls travel team from Ohio, played a Millwall Lionesses L.F.C. youth squad. And, yes, the Americans won, albeit by the slimmest of scorelines, 1:0.
The young ladies who participated in the match were not the first sportswomen to play at Wembley. That distinction had gone to members of women’s field hockey squads. [2, 3]
Attendance for the match had been projected to be around 15,000 , but in actuality it was only about 3,000.  Although thousands more also likely watched the match, as it was supposed to be televised to a “British national television audience.” 
At the time, the Cincinnati Cardinals were coached by “boat-builder and repairman” Mike Tucker , who later on became, and still is, the head coach for the women’s soccer team at the University of Dayton. 
Wembley’s usually lush pitch was in poor shape, due to an unspecified “recent rock concert,”  which likely referred to another historic event that July: Genesis’ four-in-a-row series of sold-out concerts, which ran from July 1st to July 4th. 
Information on who and how the only goal was scored is not readily available.
The Cardinals wore a red uniform that featured white accents, including a white sash, while Millwall’s kit was a white jersey with some horizontal blue stripes, blue shorts, and blue socks. 
Old Wembley was demolished in 2003, to make way for the new Wembley Stadium, but at least one other American women’s soccer squad would play on Old Wembley’s pitch. (See “Related Notes,” below.)
 That is women’s soccer match, rather than women’s soccer match. (“Women’s soccer” as in soccer played by womankind or all females, not just adult females.)
 “Schoolgirls Boot Wembley Barrier,” New York Times, July 9, 1987. (The story likely originated in a Cincinnati Enquirer or Cincinnati Post article that was picked up by a newswire, but the Enquirer‘s online paid archive does not cover 1987 and the Post discontinued publication in 2007.)
 “South Carolina soccer team completes successful camp” by Jack McCray, The News and Courier (Charleston, SC), July 19, 1987, via Google News archive (free). (This was an odds-and-ends article covering several different topics, including a Leeds United bid for Maradona and the announcement of an indoor soccer league.)
 DaytonFlyers.com biography page for Mike Tucker.
 The Waiting Room (www.twronline.net), a website for a Genesis fanzine (a fan-published magazine), specifically their “Vault” page.
 The Press Association has two photos of this match in their online public archive: reference numbers PA.7085944 and PA.7085991.
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The most famous women’s soccer match at Old Wembley has to be Italy’s 4:1 victory over England on August 18, 1990. That friendly was a “curtain-raiser” to the 1990 FA Charity Shield match between Liverpool and Manchester United. Carolina Morace scored all four Italian goals, which earned her a front-page mention in Italy’s famed La Gazzetta dello Sport (JPEG scan, www.calciodonna.it). As to the men’s match that followed, it ended 1:1, with no extra time (Wikipedia).
The first women’s (as opposed to girl’s) match involving an American team was apparently played on August 3, 1997, before the 1997 FA Charity Shield. The American squad was Penn State University’s Women’s Soccer Team, and their opponent was a select team of players from London. The select team won 3:2. (Penn State’s 2011 Women’s Soccer Yearbook; page 45, via issuu.com)
According to Penn State, Carole Dutchka was the “first-ever American woman to score a goal in Wembley.” That statement is probably true in the technical sense (woman = ‘adult female’), but unless the Cincinnati Cardinals’ goal was an own goal, then Dutchka is probably not the first American female soccer player to score a goal at Old Wembley Stadium.
And, the men’s match? Another 1:1 draw. However, Manchester United would defeat Chelsea 4:2 on penalties (Wikipedia).
The first women’s soccer match at New Wembley? It appears that there has yet to be one in the new stadium, although that should definitely change on July 31st, when Great Britain plays Brazil in their final Olympic group match. Regarding that match, a recent FA press release has this doozy of a misstatement: “This is the first time in Wembley’s history that a women’s match will take place at the iconic home of English Football.” Although, the GBR/BRA match will probably be the first official women’s international soccer match at either the old or new Wembley. (The new stadium was still under construction when England hosted the 2005 Women’s Euro, so that was not even an option.)