Rapinoe scored twice in the last 16 and again in the quarter-finals to send the U.S. to the last four but was absent from the starting lineup.
The 33-year-old, who was last week criticised by President Donald Trump for saying she would not visit the White House if the U.S. won the tournament, did not join in the warm-up before the game. She stood on the sidelines with her arms folded.
News of the injury to Bardsley, who was born in California to English parents, circulated 90 minutes before kick off in Lyon, with the Lionesses teamsheet confirming the switch.
Carly Telford, drafted in as Bardsley's replacement, featured in the 1-0 group stage win over Argentina.
Phil Neville's England side face the world champions at 1900 GMT for a place in Sunday's final.
News of Rapinoe's absence would not have pleased American fans who were already anxious before kick off.
(Reporting by Christian Radnedge; Editing by Pritha Sarkar and Ken Ferris)
