In an even affair, Asa Nicholson headed Mullets ahead just after the interval, before second-half substitute Jacob Ebling drew Wick level soon afterwards.
The decisive moment came on 80 minutes as Jenkins’ cross looped over Jordan Matthews in the Wick goal to give Mullets a derby success.
Arundel joint boss Craig Stuart admitted it was a smash-and-grab win.
“I’ve got to be honest, I thought Wick were much improved from when we last played them. They worked hard, tried to play football and were physical,” he said.
“To be fair, I think we’ve nicked it, if I’m honest. I don’t think either side deserved to win but sometimes you have to win ugly and that’s football.
“It was a bit of a smash and grab and we’ve got a little lucky.”
The game was due to be played at Mill Road but with Arundel’s pitch unplayable, the fixture was held at Crabtree Park.
Both sides cancelled each other out in the opening half, with neither creating any clear-cut chances.
Arundel needed just eight minutes to break the deadlock after the interval.
Max Thoms’ perfectly-weighted free-kick found Nicholson, whose bullet header put Arundel in front.
Wick were back in it just four minutes later thanks to substitute Ebling’s strike. Andy Chick’s cross found him at the back post and the winger’s first-time strike found the bottom corner.
Wick’s Matt Noble saw a shot tipped over by James Fernandes just past the hour before Jenkins got the winner ten minutes from time. The right-back looked to deliver a cross but the ball looped over Wick keeper Jordan Mathews and into the net.
Wick’s Kieron Playle-Howard was played in behind with five minutes to go but fired straight at Fernandes.
Wick joint boss Richard Towers felt his side deserved something on the night and said: “I don’t think we did enough to win the game but we certainly didn’t deserve to lose it.
“Their second was a bit of a fluke and a little disappointing from Jordan Matthews’ point of view but, other than that, I can’t remember him making a save.”
Despite the defeat, Wick’s safety was virtually secured with third-from-bottom Hailsham losing at home to leaders Horsham.
Towers’ side have a nine point cushion over Hailsham, who have just three games left to play.
WICK & BARNHAM: Jordan Matthews; MacIver D.Towers, Playle-Howard, Eyres; Noble, S.Towers, Joseph Matthews; Chick, Norgate, Fellows. Subs: Dunlop (S.Towers), Ebling (Joe Matthews), Forry, Tucker.
ARUNDEL: Fernandes; Jenkins, Dudas, Peake, Hunter; N.Da Costa, Jones, Thoms, Nicholson; Tipper, Herbert. Subs: L.Da Costa (N.Da Costa), Russell (Herbert), R.Biggs (Jones), O’Hagan.
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