First-half goals from Lee Tomlin and Tammy Abraham put the Robins on their way, before Aden Flint and David Cotterill added gloss to the scoreline in a match that was screened live on Sky Sports.
A look at the table after the match proved how vital the result was, with Lee Johnson’s side moving out of the bottom three and up to 19th in the table.
The Terriers, on the other hand, remain third having failed to make ground on the top two, who are six points better off.
Friday night’s other match saw two Play-Off chasers go head-to-head, with Reading beating Sheffield Wednesday 2-0 to move up to fourth in the table.
Sitting fifth at kick-off, the Royals had Yann Kermorgant and Adi Popa to thank for their goals, and they overtook Leeds United in the standings.
Table: Click here to see the Sky Bet Championship table
In Sky Bet League One, Port Vale scored two quickfire, second-half goals to overcome relegation rivals Shrewsbury Town.
After 67 minutes without a goal, Sam Foley and Ryan Taylor both found the net within three minutes to give the Valiants the upper hand.
Former Vale man Louis Dodds pulled one back for the visitors but they failed to find a leveller.
Port Vale remain the relegation zone but are two points behind both 20th-placed Oldham and Shrewsbury, in 19th, with two games in hand on those sides.
Table: Click here to see the Sky Bet League One table