And caretaker boss Sammy McIlroy was delighted by the application shown by the visitors in his first game in charge.
In truth the visitors rode their luck at times, with Steve Ferguson and Justin Richards both hitting the inside of the post as Woking searched desperately for an equaliser that never came.
Craig McAllister forced Ryan Robinson into an early save as Glenn Cockerill's men started brightly but it was Morecambe who took the lead through the lively Carlton.
Woking's left back Karim El-Salahi was robbed by Michael Twiss on the right and his angled low cross was met by Carlton.
The striker's exquisite first touch fooled Stuart Nethercott before he prodded home from close range with his second.
It was rough justice on Woking who should have equalised when McAllister flashed a free header wide.
Woking's Simon Jackson then cleared off the line as a dangerous Gary Thompson corner flew off the head of Ian Selley.
The home started the second half brightly and Ferguson rattled the post with a 20-yard effort.
Jim Bentley might have doubled Morecambe's lead but he volleyed over the bar from three yards, before Twiss fired wide as Woking left themselves slightly vulnerable at the back.
Thompson has a one-on-one brilliantly saved by Cards keeper Shwan Jalal in injury-time, before the post denied Woking a point when Richards' header thudded away to safety.
"I'm so pleased for Jim Harvey and everyone at the club," said McIlroy.