The 21-year-old, named in Stuart Pearce's party for the friendly in Italy, converted a Daniel Fox free-kick from six yards with just nine minutes left to stretch Burnley's unbeaten home run to five games.
Former Norwich loanee Dean Marney had put Burnley ahead after 33 minutes.
Following good play from winger Chris Eagles, Marney - at Carrow Road in 2005 - produced a neat finish to claim only his third goal of the season.
But Norwich equalised in the 65th minute when captain Grant Holt registered his 13th goal of the season with a fine finish from a Wes Hoolahan delivery.
The defeat ended Norwich's eight-game unbeaten run in the Championship and they have now won at Turf Moor just once in 17 league matches.
Norwich, who went into the game with the best away record in the division, made a bright start and Henri Lansbury - the injury-time match-winner against Millwall on Tuesday night - fired over the crossbar from 30 yards inside a minute.
Burnley responded and new signing Charlie Austin miscued his effort from 12 yards.
Rodriguez then glanced a header wide from similar range before Clarke Carlisle could not direct his header from a Ross Wallace corner on target.
Rodriguez fired wide and Eagles fired a 30-yard free-kick over the bar while at the other end Holt headed wide from close range.
But the home side went in front when Marney finished well after fine play from Eagles and Jack Cork then forced City goalkeeper John Ruddy into a save three minutes later.
Burnley's confidence was high and Eagles and Marney had further efforts which failed to find the target as they looked to double their advantage before half-time.
Just before the break, Norwich had half a chance but Leon Barnett headed Lansbury's free-kick over the bar.
Norwich manager Paul Lambert made a change at the break with Simon Lappin replacing Zak Whitbread, and within three minutes Holt forced Clarets goalkeeper Lee Grant into a smart save.
Holt then won City a free-kick 20 yards out and Grant pulled off a fine save to turn Lappin's effort around the post on the hour.
But the visitors hauled themselves level five minutes later when Holt finished well from Hoolahan's cross, his first goal in six matches.
Substitute Chris Iwelumo headed wide just minutes after coming on before Norwich went so close to taking the lead.
Grant made a fantastic fingertip save to turn Chris Martin's 18-yard effort around the post for a corner with 14 minutes left, and Barnett headed over from the resulting delivery.
However, it was the home side who snatched victory with nine minues left when Rodriguez slotted home.