Slovakian Vladimir Weiss impressed during the opening stages in front of a crowd of nearly 12,000, with Sturridge testing Rhys Taylor in the home goal in the 17th minute.
Four minutes later, City captain Ben Mee had a firm header turned away by Taylor, before Chelsea had a goal disallowed for handball in the build-up.
City keeper Greg Hartley, who kept his place ahead of first-choice keeper Filip Mentel, then saved well from Tejera to ensure it remained goalless at the break.
City then took the lead four minutes after the restart, with Weiss laying the ball into the path of Sturridge, who took one touch before driving home the opener.
Hartley palmed away a fierce drive from Nielsen, and City replied with David Ball being denied by Taylor before firing over from 12 yards after more good work by Weiss down the right.
Chelsea drew themselves level when Gael Kakuta headed in off the crossbar in the 66th minute.
The two sides then adopted a cautious approach as time ran out, although Chelsea almost snatched a first-leg advantage when Kakuta fired narrowly wide in the last minute.
City: Hartley, Trippier, Mee (c), McGivern, Tutte, Boyata, Weiss, Kay, Ball (Ibrahim 86), Sturridge, McDermott (Tsiaklis 72).
Unused subs: Mentel (GK), Poole, Nimely-Tchuimeni.