Substitute Jackson took advantage of an error by West Brom defender Craig Dawson after Marc-Antoine Fortune had brought the hosts back on level terms.
Norwich captain Grant Holt's ninth goal of the campaign had separated the sides at half-time.
But the real Norwich hero was 19-year-old debutant goalkeeper Jed Steer who made several fine saves.
His efforts enabled Norwich to complete a quickfire double after their Premier League win by the same scoreline at The Hawthorns a fortnight ago.
West Brom did the initial pressing in this contest but it was 21 minutes before the first meaningful action of the game.
Steer did well to turn a fierce half-volley from Jerome Thomas against the bar from a Gonzalo Jara Reyes centre.
James Morrison linked up well with Graham Dorrans on the edge of the Norwich box but the home captain dragged his eventual shot wide.
West Brom were looking the more likely to break the deadlock and Fortune was just too high with a near post glancing header from a Dorrans corner.
Thomas continued to pose a threat but was yellow-carded after half an hour for a mistimed challenge on Russell Martin.
In a rare Norwich attack, Dawson produced a fine tackle on Grant Holt inside the box as he prepared to shoot.
But after 35 minutes Holt put the Canaries in front from a rapid counter-attack.
Wes Hoolahan played in Elliott Bennett who lost marker Joe Mattock down the right, and his low centre was turned home at close range by the alert Holt.
The visitors took confidence from the breakthrough and Ben Foster had to deal with Whitbread's downward header from a David Fox corner.
Hodgson opted for a half-time switch, with central defender Gareth McAuley replacing Jara Reyes.
He also moved Cox into a central attacking role alongside Fortune.
The pair quickly linked up, with Cox having a shot deflected over the bar from Fortune's low centre.
Then after 54 minutes Fortune brought Albion level.
Thomas was the creator, skipping past the challenge of Russell Martin before getting to the byline and picking out the unmarked Fortune who drilled his shot past Steer.
Holt became the first Norwich player to be booked, for a challenge on Youssouf Mulumbu after an hour.
Martin followed him into referee Martin Oliver's notebook for bringing down Fortune.
West Brom continued to dominate and Dawson's shot was only just too high from a quickly-taken Dorrans free-kick.
Foster had to be alert to deal with a deflected Fox drive in a rare Norwich attack.
West Brom substitute Shane Long was instantly brought down by Daniel Ayala who was booked for the challenge.
With five minutes left a mistake by Dawson presented Jackson with the winner.
Dawson was dispossessed inside his own box and Jackson needed no second invitation to drill a low shot past Foster into the far corner of the net.
Steer turned aside a Long header to prevent an equaliser with a minute remaining. Then in injury-time he repeated his heroics to block a shot from Mattock.