The home side battled bravely despite being a man down for the entire second period following Christopher Schindler's dismissal for two yellow-card offences just before the break.
— Premier League (@premierleague) January 2, 2019
Burnley also ended with 10 men after substitute Robbie Brady's straight red card for a cynical challenge on Isaac Mbenza in the closing stages.
The Clarets' first league win on the road since the end of September lifted them up to 16th in the table and crucially dealt Huddersfield's survival chances a major setback, leaving them rooted to the bottom, eight points from safety.
Huddersfield created the game's first half-chances. Philip Billing miscued his shot from the edge of the penalty area and Mounie's overhead kick cleared the crossbar.
Mounie's low shot after a surging run was off target and it was not until the 20th minute when Burnley first threatened, with teenager Dwight McNeil failing to pick out a team-mate after snaking to the by-line.
McNeil fired a low effort just wide soon after and Ben Mee's header forced a fine save from Huddersfield goalkeeper Jonas Lossl, who reacted sharply to block Barnes' follow up.
At the other end, Phil Bardsley headed Elias Kachunga's thumping goalbound header clear before the Terriers broke the deadlock in the 33rd minute.
Mbenza, back in the starting line-up, curled over an excellent cross and Mounie headed home his first goal of the season.
It was the first goal scored by a Town striker this season and only the club's fifth at the John Smith's Stadium during the current campaign.
But Town's ascendancy was short-lived. McNeil ghosted past Florent Hadergjonaj on the left edge of the area and his low cross was turned home by Wood in the 40th minute.
The home side were then dealt a hammer blow just before the interval when Schindler was shown his second yellow card by referee Mike Dean for his challenge on McNeil.
The German central defender was reluctant to leave the field, apparently unaware he had already received an earlier booking for his foul on Wood in the 30th minute.
Wood headed Johann Berg Gudmundsson's header narrowly wide at the start of the second period and the latter fired straight at Lossl following a neat interchange on the break with Ashley Westwood.
Gudmundsson curled an effort wide with 25 minutes left as 10-man Huddersfield's endeavours were appearing to take their toll and substitute Brady, on for the impressive McNeil, shot straight at Lossl.
But Burnley carved the home side open for their 74th-minute winner.
Gudmundsson, Wood and Westwood combined to set up Barnes inside the area and the Clarets striker made no mistake from 10 yards.
Town defender Terence Kongolo headed Brady's dangerous cross out for a corner late on before the Irishman saw red for his professional foul on Mbenza.
Billing curled the subsequent free-kick narrowly off target as the Terriers went down fighting.