Winger Schofield got no joy down the left wing from Wales international Sam Ricketts in the opening 20 minutes, but swapping sides paid immediate dividends.
Schofield twice beat Alan Tate down the right to supply crosses that allowed Andy Booth and Pawel Abbott to score from close range on 27 and 41 minutes respectively.
Swansea striker Lee Trundle cancelled out Booth's opener with a penalty on 32 minutes awarded after he was tripped in the box by Terriers defender Nathan Clarke.
Abbott put Town back in front before the break with his third goal in as many matches and Schofield then capped a fine afternoon's work with a long-range drive after the restart that was deflected as it beat goalkeeper Willy Gueret
Swansea's misery continued when stalwart defender Christian O'Leary was sent off for elbowing Jon Worthington in the face
The game featured eight bookings in total - three for Huddersfield and five for the visitors as it threatened to boil over
Both Trundle and Huddersfield defender Danny Adams were lucky to receive only yellow cards for an incident in first-half stoppage time when the Swansea striker appeared to elbow Adams in the face with the defender pushing and shoving Trundle in return.
It got worse for Swansea in the closing stages when Izzy Iriekpen, who otherwise had a good game, was given a straight red for violent conduct committed right in front of York-based referee Matt Messias.
The action-packed contest entertained both sets of fans with plenty of goalmouth action.
Swansea nearly went in front with a brilliant free-kick routine on 17 minutes that forced Paul Rachubka into an outstanding double save.
In the second half Hudderfield nearly stretched their lead late on when Chris Brandon robbed Tate inside his own half and dashed 60 yards but blasted wide of the target after out-pacing the entire Swansea defence.