The score was level at half-time and it was Dominic Solanke who broke the deadlock just two minutes into the second half.
Solanke was played through by Lewis Cook and fired it past Mexico's Abraham Romero to give England the lead.
England had a flurry of chances to double the lead in the second period and it was Josh Onomah who came close with a curling effort that was tipped on to the post.
Paul Simpson's side had to hold on with 10 men, with Onomah being shown a second yellow card after standing on the ankle of Juan Aguayo.
The Young Lions could have rounded it off with a little over 10 minutes remaining when Solanke charged down on goal before laying it off to Dominic Calvert-Lewin, whose shot was blocked by Romero.
Fry played the full game, while Harry Chapman had to settle for a place on the bench.
England will face Italy in the semi-finals on Thursday, June 8 (8pm).