France made it three wins out of three in the group's other game - hammering England 3-0 to head to the knockout stages as the only team with a perfect record.
The point shared meant Spain finished second in Group C on four points behind leaders France on nine.
They joined Sweden and Italy, Group A winners and runners-up respectively, and Norway and Germany from Group B, in the last eight.
Iceland, with four points in Group B, qualified for the knockout stage as one of the best third-placed teams.
The last quarter-final berth will be filled by Denmark, who were third in their groups with two points each.
Russia lost out in the draw with the Denmark assistant coach Soren Randa-Boldt commenting: "I we deserved it. I think we were a better team than Russia. "
Sweden v Iceland - Sunday
Italy v Germany - Sunday
Norway v Spain - Monday
France v Denmark - Monday