Swansea extended their unbeaten away run to nine Sky Bet Championship matches but squandered the lead in a 1-1 draw at 10-man Huddersfield.
While results at home have become an issue for Steve Cooper's men, winning only one match – against rivals Cardiff – at the Liberty Stadium since August, the Swans have fared considerably better on the road.
Midfielder Fulton's thumping opener for Swansea came against the run of play in the 18th minute, but Grant had Huddersfield level four minutes from half-time with a cool finish.
The visitors, who made four changes from Saturday's 1-0 defeat to Millwall, came under pressure early on, with goalkeeper Freddie Woodman forced to tip a 25-yard Chalobah effort over his crossbar inside two minutes.
Jonathan Hogg earned the first yellow card of a feisty opening 45 minutes for a clumsy challenge on Connor Roberts and things soon got worse for the hosts, who conceded from the first real period of Swansea pressure.
Hogg's attempted clearance from a corner was headed back into the danger area by Ben Wilmot, with Fulton firing a powerful finish into the roof of the net after the ball was poked to him in space to the right of the goal.
Home fans voiced their growing dismay at the performance of referee Jarred Gillett as the booking of Fraizer Campbell for a heavy challenge on Mike Van Der Hoorn required lengthy deliberation, but they were celebrating in the 41st minute.
Alex Pritchard's shot from just outside the box deflected kindly into the path of Grant, who kept his composure to place a shot beyond Woodman for his 10th goal of the season.
Van Der Hoorn did not emerge for the second half, replaced by Ben Cabango, whose first contribution on his league debut was an important interception to snuff out a threatening left-wing cross.
Woodman touched over a Pritchard free-kick just short of the hour mark and Juninho Bacuna then flashed a shot across goal from a tight angle.
Pritchard again tested Woodman with a low strike on the turn in the 72nd minute as the Swansea defence continued to come under pressure, but Huddersfield's hunt for a winner was dealt a setback when Chalobah saw red four minutes from time.