Gray followed up a Kyle Lafferty shot for the opener and added his second from short range on 40 minutes.
Mike Proctor reduced the deficit after the break, then equalised from the spot after Eifion Williams won a penalty.
But Joey Gudjonsson immediately fired in a vicious shot, before Ade Akinbiyi and Rodriguez rounded off the victory.
Wrexham: Michael Jones, Ryan Valentine, Shaun Pejic, Richard Hope, Neil Taylor (Mike Williams 57), Mike Carvill(Matty Done 57), Levi Mackin, Conall Murtagh (Mark Jones 60), Eifion Williams (Chris Llewellyn 75), Kaid Mohammed (Marc Williams 60), Michael Proctor.
Subs: Anthony Williams, Gareth Evans.
Burnley: Brian Jensen (Gabor Kiraly 46), Stephen Foster, Michael Duff, Steven Caldwell, Jon Harley, Besart Berisha (Stephen Jordan 64), James O'Connor (Alan Mahon 46), Jon Spicer (Joey Gudjonsson 46), Steve Jones, Andy Gray (Ade Akinbiyi 46), Kyle Lafferty (Robbie Blake 26) (Jay Rodriguez 80).
Referee: Mike Jones.
Att: 1,457
BBC