The visitors took advantage of a below-par Wycombe display to lead 1-0 at the break courtesy of Klukowski's crisp finish, but a vastly improved performance after the break saw the hosts level through Fred Onyedinma's close-range effort.
However, a lapse in concentration from a set piece allowed Aaron O'Connor to lay it on a plate for Klukowski to snatch the winner, ending Wanderers' unbeaten record on Saturday fixtures this season.
Gareth Ainsworth shuffled his pack after the midweek defeat to Plymouth and made three changes in personnel while switching to a 4-4-2 formation, with Steven Craig and Matt McClure leading the attack while Peter Murphy returned in midfield.
There was very little goalmouth action in the opening 15 minutes with both sides finding their feet and Joe Jacobson’s free-kick hit the wall in the first attempt of the game.
So it caught most fans in the ground by surprise when Yan Klukowski drilled home the opener from just inside the box after a cross from the right found its way to him, and the former Forest Green man hit a clean first-time shot into the bottom corner to make it 1-0.
Wycombe struggled to create much at the other end on a bobbly surface and the best endeavours of McClure and Craig may have unsettled the defence but failed to result in opportunities to test the keeper.
Both strikers fired over the bar before Darren Jones headed wide from a corner for County, and then Matt Ingram was called upon to make a vital save before the break when the hosts failed to clear on the edge of their box and Mark Byrne poked goalwards, only for the keeper to block his effort six yards out.
Then Matt McClure had a golden chance to level the scores when he latched on to a poor backpass but attempted to take the ball round the keeper rather than get his shot away, and Joe Day stuck out a hand to scramble it clear.
The half-time interval brought no changes for either side but Wycombe seemed to have more impetus and saw Matt McClure’s near-post effort saved after he shot first-time as the ball landed over his shoulder.
And they were level within 10 minutes of the restart when Sam Wood dug out a powerful shot from 25 yards and although Day made a save down to his left, the ball fell into the path of Fred Onyedinma who made no mistake in tucking it home.
All of a sudden the hosts grabbed the game by the scruff of its neck and got well on top, seeing Alfie Mawson’s header excellently saved by Day at the near post, before substitute Paul Hayes drilled a shot straight at Day after good hold-up play by Onyedinma.
A winning goal looked only a matter of time for the home side as they played with confidence but an injury to Joe Jacobson just took the sting out of the tempo and County regained the lead totally against the run of play with 11 minutes remaining.
Wycombe weren’t happy about the award of a free-kick on the left wing and although the delivery missed everyone, Aaron O’Connor returned it into the danger zone and Klukowski was on hand to drill home from 12 yards to stun the home side.
Aaron Holloway was summoned from the bench in a bid to rescue a point against his former side and his header forced a save from Joe Day in the closing stages, before Matt Ingram prevented Shaun Jeffers from stretching the lead in stoppage time.
Ainsworth's men couldn't summon an equaliser and will hope their excellent away record comes into play again next week for the long trip north to Carlisle.
Wycombe: Ingram, Jombati, Jacobson, Mawson, Pierre, Bean, Murphy, Wood, Onyedinma, McClure (Hayes 66), Craig (Holloway 86).
Unused subs: Richardson, Rowe, Fletcher, Bloomfield, Ephraim.
Newport: Day, Jackson, Tutonda, Jones, Yakubu, Byrne, Porter, Klukowski, Storey (Chapman 76), Howe (Feely 86), O’Connor (Jeffers 80).
Unused subs: Stephens, Flynn, Parker, Poole.
Winning 50/50 Number: 137449
Prize Money: £621.00