Liverpool midfielder Shelvey's first goal for the Young Lions followed Steven Caulker's third to send Stuart Pearce's men five points clear at the top of qualifying Group Eight with one match remaining.
That is on Monday against Norway, who won 3-1 in Belgium in Thursday's late kick-off. In Chesterfield in four days' time, the Norwegians could deny their rivals a place in next month's two-legged play-offs.
Thursday's victory was the Young Lions' sixth out of their seven qualifiers and saw Pearce hand a debut to Thomas Ince, son of former England star Paul, on an artificial pitch against opponents England had thrashed 6-0 a year earlier.
And after Javid Imamverdiyev went close for the hosts, in-form Blackpool winger Ince should have stolen the headlines when he screwed Marvin Sordell's cross wide on the turn.
England assumed total control and some delightful football saw Wilfried Zaha bamboozle the Azeri defence and fire into the side netting.
A series of corners then came to nought before Craig Dawson was booked in the 25th minute for exacting retribution on Rizvan Umarov after being caught himself.
But Dawson had the last laugh three minutes later when centre-back partner Caulker powered England ahead, the Tottenham man heading in Shelvey's left-wing free-kick.
Caulker should have made it 2-0 two minutes later when he glanced Ince's far-post cross wide before Azerbaijan goalkeeper Salahat Agayev produced some unconventional heroics to keep the hosts in it.
A goalmouth scramble saw him keep out Martin Kelly's deflected cross and deny Marvin Sordell before Zaha's follow-up was cleared off the line to muted penalty appeals.
Agayev then looked stranded after completely missing a cross but he recovered to produce an astonishing near-post save from Sordell, who looked certain to convert after Danny Rose kept the ball in play.
Caulker saw another header go to waste shortly before half-time and Zaha fired a good chance over shortly after the restart.
Azerbaijan, who had thrown on Orkhan Hasanov at the break, began to get a grip and, after some harmless long shots, they might have levelled on the hour mark when Imamverdiyev curled narrowly high and wide.
Pearce took immediate action by sending on debutant Ben Marshall for Zaha but the hosts continued to push.
Ben Amos - who had looked untroubled filling in for ill goalkeeper Jason Steele - saved from Araz Abdullayev, prompting Connor Wickham's introduction for Sordell.
The second change almost proved inspired as Wickham nodded Martin Kelly's cross wide but it appeared to have the desired effect as England kept their opponents at bay.
Mirhuseyin Seyidov's free-kick was easily claimed by Amos and Azerbaijan went for broke in the final 10 minutes with a double substitution of their own.
But it backfired seven minutes from time as Wickham nodded Caulker's long ball into the path of Shelvey, who volleyed emphatically beyond the onrushing keeper.