The Canaries were trailing 1-0 when Elliott entered the fray in the 62nd minute but by the end of the match League One leaders City had sealed a fourth straight win and equalled a club record with their 10th away win of the season.
Defender Neal Trotman headed in Anthony Pilkington's corner at the near post after only three minutes to give Huddersfield the lead and Town went on to dominate for much of the game.
The Terriers played some superb football, only for moves to break down or City keeper Fraser Forster to deny them.
Huddersfield were made to pay the price in the 69th minute when Grant Holt side-footed in his 27th goal of the season from Wes Hoolahan's cross.
The home side then capitulated, with Elliott putting Norwich in front with a tap-in from Holt's pull back and sealing victory with a 20-yard drive five minutes from time.