Only four minutes after coming on as a substitute, Cain collected a superb 68th-minute pass from Rachel Rowe and rounded goalkeeper Jennifer Falk before finishing via a post.
Sweden had taken the lead through Magdalena Eriksson on the hour and while the Group A4 pacesetters dominated the closing stages, they were unable to find a winner.
Wales were missing veteran duo Jess Fishlock and Sophie Ingle and had made six changes to their starting line-up, but they still showed resilience late on with keeper Safia Middleton-Patel making several crucial saves.
It is the second time they have drawn 1-1 with Sweden having held the group favourites to the same scoreline in Wrexham in February. Wales remain bottom of the group on two points.