The Owls were a man down from the 32nd minute when Barry Bannan was sent off for bringing down Jason Knight from behind as he prepared to shoot. City claimed a penalty, but referee Lewis Smith ruled that the offence was just outside the box.
Wednesday defended valiantly, but were undone after 64 minutes when centre-back Dickie slid in to convert Tommy Conway’s deflected cross from the right.
It meant a positive start to life without the sacked Nigel Pearson for the home side, but they were made to work hard by determined opponents belying their lowly status in the table.
City made a bright start, with Conway shooting narrowly wide and Knight firing straight at goalkeeper Cameron Dawson.
Wednesday took time to grow into the game, but created a great chance after 24 minutes when Anthony Musaba headed over from a Reece James cross.
Musaba shot over before Bannan received his marching orders for fouling Knight. From the resulting free-kick, the ball broke to George Tanner, whose low shot was palmed around a post by Dawson.
Taylor Gardner-Hickman shot over for City, but Wednesday defended with tenacity until the break, Bambo Diaby and Dominic Iorfa excelling in a strong back-line.
Visiting boss Danny Rohl made two changes at the break, sending on Pol Valentin and Will Vaulks for Michael Smith and Djeidi Gassama.
Wednesday began the second half on the front foot and City goalkeeper Max O’Leary had to be alert to palm out Iorfa’s far-post header from a left-wing corner.
Cameron Pring had a 25-yard shot saved by the diving Dawson, but the hosts were finding it difficult to create openings. By the hour mark, Fleming had seen enough and made a triple substitution – Kal Naismith, Sam Bell and Anis Mehmeti replacing Zak Vyner, Tanner and Andreas Weimann.
City began to dominate after Dickie’s goal and should have gone two up after 72 minutes. Conway’s low shot came back off a post and from the rebound Bell somehow contrived to hit the other upright with the goal at his mercy.
Instead of being out of sight, the Robins had to endure a nervy finish and were grateful to O’Leary for a brilliant save near the end, keeping out substitute Ashley Fletcher’s close-range shot.
City had another chance in stoppage time when goalkeeper Dawson went forward for a corner and Conway raced away on the break only to delay his shot and allow a recovery tackle.