Crofts scored his seventh and eighth goals of the season with a glancing header and a stunning volley in the first half of the Championship clash in south Yorkshire.
The victory sees Norwich strengthen their claim to a play-off position and continue their excellent away record, having only lost three fixtures on the road this season.
The Canaries started brightly, Wes Hoolahan beginning his 100th league appearance for the Norfolk side with a powerful shot from distance which Luke Steele did well to parry.
The home side struggled to close the Republic of Ireland international down for much of the first half, and he was on hand to test their goal again after 16 minutes with a fizzing shot from 30 yards which Steele had to beat away.
Barnsley sparked into life soon afterwards, Goran Lovre finding space to force a splendid reaction save from John Ruddy with a bullet header.
The first blow in a miserable afternoon for the hosts came when Bobby Hassell was taken off injured, replaced by new loan signing Marlon Harewood.
After a brief respite, Paul Lambert's men initiated another swift attack, skipper Grant Holt stabbing a shot into the side-netting on the half hour mark.
Two minutes later, Marc Tierney was allowed the freedom of Oakwell to switch the ball to David Fox who clipped the ball onto Crofts' head for him to glance an opener with his forehead.
Things went from bad for worse for Mark Robins, managing against his former club, when fellow former Canary Jason Shackell cut his lip in a clash of heads.
As he received treatment on the sidelines, the extra space afforded to the Canaries resulted in Crofts' second goal.
Hoolahan breezed through the red shirts to tee up a clever ball for Crofts, who manoeuvred himself perfectly to volley left-footed past Steele with 40 minutes played.
Shackell was replaced by Jim O'Brien before half-time after his injury proved too severe to play through.
After the restart, Kieran Trippier produced possibly Barnsley's best chance of the encounter, crossing into the onrushing Harewood's path for the debutant to clip his shot over the crossbar.
Crofts was close to scoring a perfect hat-trick but his drilled right-footed shot bobbled just past the upright with 25 minutes remaining.
The visitors were unlucky not to score a third and add extra gloss to the victory when Steele fumbled a loose ball in his area on 80 minutes after a Holt snapshot.
The Tykes lacked cutting edge in the final third throughout and fell to their first defeat in four home matches as the final whistle sounded.