Sidley’s Scott McDonald’s 82nd minute spot-kick for handball against Harry Street was parried by keeper Matt Short, who did well to block the rebound.
Short was then the hero at the other end when he lashed home a stoppage-time penalty winner following Dan Cruickshank’s challenge on Michael Turvey,.
City boss Adie Girdler said: “It was very pleasing. We didn’t play well in the first half but in the second, we were the better side.
“It was Sam Doran’s first game since his honeymoon. He came in at left-back and had an excellent game, and marshalled the defence.
“Archibald did very well, Bass was excellent in midfield and Lewis Edwards, who we played at the top, looked a bit livelier. And Matt Short was excellent.”
Girdler is not getting carried away by a win which earns City a Vase first round proper trip to Molesey on October 22 as he remains focused on an injury crisis.
Striker Ben Vassallo heads the list of casualties, but Girdler is confident he will have a replacement by Saturday, when they travel to Little Common in the Sussex Senior Cup.
At Sidley the home team wasted numerous chances before City took the lead midway through the first half when Sidley failed to deal with the second ball following a free-kick and Phil Archibald put a volley past Josh Willis. Soon after, Willis denied the impressive Edwards from a set-piece.
Scott McDonald got the Blues back on level terms in the 37th minute with a sharp finish.
Right on half-time Marc Munday got up at a Jason Taylor corner to power home a header.
Chances were scarcer after the break, but Harry Bass equalised with a back-post header before the late penalty drama at both ends.
CITY: Short, James (Stuart 28), Doran, Street, Wallis, Bass, Mattison, Archibald, Turvey, Edwards, Cotton (Groom 42). Sub not used: Gilbert.
Chichester lost 2-1 at Wick in the John O’Hara Cup on Tuesday.
SIMON CAREY & JACK PRESTON