The striker replaced Tom Ince in the 79th minute at Swansea's Liberty Stadium but picked up the injury - a sprained ankle - just minutes later after a challenge with Wales defender George Ray. He was replaced by Jake Foster-Caskey in the 84th minute.
England coach Gareth Southgate confirmed he would now miss out on the Toulon Tournament.
"It’s a real shame because Harry has been extremely patient for his chance and I was looking forward giving him some of the prestigious games in Toulon," Southgate told thefa.com.
"He’s warranted that opportunity and has had a super finish to the season with Tottenham so it’s disappointing for us and for him."
England beat Wales 3-1 thanks to a hat-trick from Norwich's Nathan Redmond to further strengthen their position at the top of Group One in qualifying for the 2015 European U21 Championships.