They then meet Belgium at Barnsley's Oakwell stadium, as they seek to secure a spot in the final qualifying round for next summer's championship.
It means Cork, currently on loan at Scunthorpe from Premier League side Chelsea, is likely to get his first real taste of competitive action in North Lincolnshire wearing white rather than claret and blue.
Although the first-half at the Keepmoat Stadium in Doncaster was goalless, England stepped up a level after the restart.
Manchester United youngster Febien Brendy put Eastwick's side ahead just 60 seconds into the second period.
And although the visitors levelled after 53 minutes through Aron Einar Gunnarsson's header, England fired in four further goals to claim the win.
Newcastle striker Andy Carroll reclaimed the lead when his shot was deflected home following Sam Hutchinson's cross.
James Henry made it 3-1 after 61 minutes with a stunning long-range effort.
Krystian Pearce was allowed the freedom of the pitch to head in England's fourth, and Scott Sinclair capped off the win with a fine solo goal.