The Terriers frontman saw his first spot-kick saved on 61 minutes after a long delay for a red card mix-up by referee Richard Clark.
The delay was due to the official sending off Cauley Woodrow for deliberate handball, who refused to leave the pitch and had to be dragged to the touchline by his team-mates.
Further protests saw Clark consult the fourth official and Fulham’s bench and led him to rescind the red card.
Clark then sent off Shaun Hutchinson for the handball with Wells finally taking – and missing – the penalty after a five-minute delay.
It was the second red card controversy of the day after Neil Swarbrick sent off West Brom’s Gareth McAuley instead of Craig Dawson in the match at Manchester City. That mistake – for which Swarbrick later apologised – was not corrected during the game.
Wells had the chance to make amends from the spot again 10 minutes later after Sean Scannell was felled in the box by Ashley Richards – who was booked for the challenge – on 71 minutes.
However the former Bradford striker, in his first appearance in five matches due to a shoulder injury, hit the post from the spot.
All this happened with the home side already behind from a bizarre goal which Fulham scored in the second minute, direct from a corner.
Alex Kacaniklic hit a low effort without any pace to the near post from the right and it somehow crept in.
Huddersfield keeper Alex Smithies and Mark Hudson were guarding the near post but both seemed to freeze, possibly distracted by the run from deep to the near post by Cottagers striker Matt Smith.
Town had further penalty appeals in both halves turned down on a day when they did everything but score and dominated the game.
A cross from Huddersfield’s Ishmael Miller on 40 minutes twice hit the post after it flicked off Wells.
The ball first struck the inside of the left-hand upright, hit keeper Marcus Bettinelli, bounced back onto the post and was then cleared.
Fulham had not won any of their previous eight away matches, losing six, and had won only one of their last 13 outings.
Huddersfield’s afternoon was summed up on 84 minutes when Jacob Butterfield hit the bar with a 20-yard drive.
Scannell drew a good save from Bettinelli late on and then in the sixth minute of stoppage time Fulham landed a killer blow on the break.
Substitute Seko Fofana dashed from inside his own half and finished at the second attempt after hitting the post.
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