The Lionesses had the opportunity to at least force extra-time with an 84th-minute spot-kick, but Houghton's poor effort was saved and Millie Bright saw red late on as USA held on to ensure their title defence goes to the final stage.
Co-captain Rapinoe's absence - said to be due to a hamstring issue - provided a pre-match shock, but Christian Press made a swift impact as her replacement when heading in the opener, before Ellen White's equaliser was cancelled out by Morgan nodding in just past the half-hour mark on her birthday.
England produced a good response in the second half and White had a second goal ruled out for straying marginally offside, before Alyssa Naeher easily saved Houghton's penalty and a frustrated Bright earned a second booking, providing a disappointing end for Phil Neville's side.
A breathless start resulted in USA opening the scoring early, with Press meeting Kelley O'Hara's deep cross with a fine header into the top-left corner.
But England – who had looked overwhelmed – leveled nine minutes later, White expertly guiding Beth Mead's left-wing cross into the top-right corner with a first-time effort.
Morgan ensured USA went into the break ahead, however, racing on to Lindsey Horan's lofted pass and glancing a header beyond Carly Telford, before appearing to mock England in her celebration, as she pretended to sip from a cup of tea.
The Lionesses improved after the break and thought they had the equalizer 23 minutes from time as White converted after Jill Scott's gorgeous flicked pass, but a VAR review deemed the striker to have been just offside.
Another VAR referral nine minutes from time gave England a penalty after Becky Sauerbrunn was deemed to have fouled White with the goal at her mercy, but Houghton's feeble effort, taken a full three minutes later, was easily saved.
Bright then showed her annoyance with a strong tackle on Morgan that earned her a second yellow card, and USA subsequently finished the job.
