Academy product Clarke headed Tony Dinning's left wing cross past Terriers goalkeeper Paul Rachubka after fellow defender Efe Sodje failed to clear.
The effort was officially credited to Clarke but Robins substitute Leroy Lita claimed in a post match interview that the ball flicked off the back of his head before crossing the line and should therefore be awarded to him.
It was not the only disputed own goal in an entertaining contest. Dinning gave Huddersfield the lead after 18 minutes when he sliced an attempted clearance past visiting keeper Steve Phillips from close-range.
Video replays showed the ball might not have gone in without a final touch off City defender Clayton Fortune but either way it went down as an own goal.
Another defensive blunder put Huddersfield two up early in the second half.
Robins captain Danny Coles tried to head a long punt back to Phillips but let Pawel Abbott in to register his 17th goal of the season with a cheeky lob over the advancing keeper.
The action-packed match featured five bookings, Huddersfield's Chris Brandon hit the woodwork with a long-range effort and both sides had goals ruled out.
At 2-0 down City seemed to be heading for defeat but player-manager Brian Tinnion produced a masterstroke by bringing on Lita and switching his team's formation from 4-4-2 to 4-3-3.
Within minutes of the changes Lita brought City back into the match with a close-range diving header and could have levelled soon after with a terrific long-range effort that was cleared off the line by Town defender Andy Holdsworth.
Lita was again involved in the move that finally brought City level - the second time this season that the Robins have clawed back a two-goal deficit against Huddersfield.
City were 3-1 down at Ashton Gate in September but two goals in the last nine minutes ensured that encounter ended all square as well.