The win was Exeter's first success at home since the end of November.
Wednesday had the best chance of an even first half when Miguel Angel Llera headed a corner goalwards.
The ball seemed destined for the top corner, but Exeter goalkeeper Artur Krysiak threw out his right hand to make a superb save, while Exeter's Daniel Nardiello had a goal ruled out for offside.
Wednesday opened the scoring in the 65th minute when a long throw-in was cleared as far as Jermaine Johnson and he lashed the ball into the bottom corner from 25 yards.
But Exeter hit straight back as Jones curled a superb free-kick round the Wednesday wall and into the bottom corner after a foul on Daniel Nardiello.
With six minutes remaining, the Grecians went in front through David Noble's stunning free-kick. The ball seemed to be heading way over the crossbar, but it dipped and hit the upright before nestling in the back of the net.