Ragnar Sigurdsson got them back on level terms (six) before a mistake by Joe Hart allowed a tame shot from Kolbeinn Sigthorsson to put the minnows in front (18).
Defeat in such ignominious circumstances means the inquest into their elimination will now begin, with manager Hodgson confirming his resignation minutes after the final whistle.
Meanwhile, Iceland's fairytale tournament continues and they can look forward to a quarter-final meeting with host nation France at the Stade de France on Sunday.
Hodgson made six changes from the goalless draw against Slovakia, opting to recall Raheem Sterling, and the winger earned England an early penalty, latching on to Daniel Sturridge's pass before tumbling under the challenge of Halldorsson.
And Rooney, on the occasion he equalled David Beckham's record of 115 England caps for an outfield player, made no mistake from 12 yards.
England were ahead for just 84 seconds as an Aron Gunnarsson long throw was flicked on by Kari Arnason and turned home by Ragnar Sigurdsson.
It got better for Iceland 12 minutes later when Sigthorsson slotted his side in front, leaving England stunned. The Nantes striker was allowed too much space on the edge of the box but Hart, who got a hand to the shot, allowed it to squirm over the line.
England dominated possession as in previous matches but once again struggled to carve out chances.
A volley from Kane forced Halldorsson into a smart save while Chris Smalling headed over from close range at the back post, but Iceland's stubborn rear-guard comfortably saw in half-time.
Jack Wilshere and Jamie Vardy were introduced after the break but Iceland created the second half's first big opportunity when Hart denied Ragnar Sigurdsson's overhead kick from point-blank range.
Dele Alli misfired from 12 yards as England's frustrations mounted and time ticked away; Iceland succeeding in forcing their opponents into hopeful shots from distance.
A cute pass from Wilshere presented Kane with his clearest opportunity with 11 minutes to play, but the Tottenham striker placed his header straight at Halldorsson.
Gunnarsson had the chance to exacerbate England's embarrassment, breaking through on goal and holding off the challenge of Wilshere, but the Cardiff City midfielder saw his effort palmed clear by Hart.
Vardy almost nabbed a dramatic leveller in stoppage time but fantastic defending by Arnason re-directed the striker's header out for a corner.
The final whistle was greeted by jubilant celebrations by the Iceland players and staff, while England's players were left dejected and Hodgson headed straight down the tunnel.
Comments