Sarina Wiegman's Lionesses knew they would secure a top-two finish in Group A3 and qualify automatically for the tournament next July in Switzerland if they avoided defeat.
England created few chances in Gothenburg but clung on despite late pressure from 1984 winners Sweden, who would have qualified themselves with a victory.
"I'm happy, I'm very relieved because it was a tough game," coach Wiegman told ITV. "The first half I think we dominated.
"When you are so dominant we have to create more chances and score goals. In the second half we couldn't keep up that level.
"We did fight for it. I think we can do a little bit better, but keeping the 0-0, qualifying in this group, I am relieved."
The Scandinavians finished third in the group and will have to settle for a place in the play-offs to be played later this year.
Elsewhere, Italy and the Netherlands progressed from Group A1, in which all four teams still had a chance of going through ahead of Tuesday's fixtures.
Italy romped to a 4-0 victory over Finland to finish top, while the Dutch secured second place with a 1-1 draw against Norway.
