Goals from Marvin Sordell, Martin Kelly, Craig Dawson and a late brace from Gary Gardner did the damage for Stuart Pearce's team, who sit pretty on top of Group 8 with four wins from their opening four games.
Ameobi wasted little time in making an impression. The leggy winger beat his man on 13 minutes and sent over a deflected cross that caught Jordan Henderson by surprise. Had he seen it coming, the Liverpool player would have scored.
Soon after Henderson sent Sordell away clean on goal. The striker composed himself, but his left-footed shot from 15 yards was well saved by Arnar Peturrson in the Iceland goal.
England were getting closer. On 19 minutes Oxlade-Chamberlain caught the Iceland defence napping and nipped in on the by-line. The Arsenal player tried a cute chip from a narrow angle, but missed the target.
On 34 minutes, Sordell's strong run and cross looked to release Jason Lowe, making a run from deep. But Peturrson was alert the danger and came quickly to claim.
An England goal was beginning to seem inevitable, and when it did arrive on 39 minutes it was no surprise the impressive Sordell delivered it. Having sent Henderson away on the right he arced a run into the box, and met the midfielder's cross with a composed finish from 10 yards to put the home side ahead.
It was no more than Pearce's team deserved and they rightly went into the break with a 1-0 lead.
The second half saw England's dominance continue. Dawson stabbed home from a corner on 48 minutes, but the whistle had already gone for offside.
Soon after Iceland had a rare glimpse of goal themselves, when Johann Gudmundsson's superb free-kick from 25 yards clipped the frame of the goal.
But the visitor's hopes were ended when Kelly made an interception on the halfway line, broke through another challenge and blasted a shot home to put England 2-0 up, just shy of the hour mark.
In the 86th minute Dawson made it 3-0 with a firm header from Henderson's right-wing corner.
Substitute Gardner added a fourth with a superb curling free-kick, then helped himself to a second in the space of two minutes with a 20-yard drive after being teed up by Oxlade-Chamberlain.
England's late flourish made it 5-0 and kept Pearce's team very much on course for Israel 2013.