The home side went a man down after 36 minutes when midfielder Adam Randell was shown a second yellow card for a foul to prevent a counter-attack.
Five minutes later Coburn converted a low Ryan Leonard cross from the right with a near post volley.
Coburn then doubled the advantage after 50 minutes, taking advantage of wretched defending to net unchallenged with a downward header from 10 yards.
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The result meant that Michael Skubala's reign as City boss has begun with two home defeats, following the shock Carabao Cup exit at the hands of League Two Walsall, and without his side scoring a goal.
Skubala brought in Rob Atkinson at the back in place of the injured Cam Pring, along with attackers Dom Ballard and Yu Hirakawa, while Alex Neil gave Millwall league debuts to Tairyk Arconte and Jenson Metcalfe.
The visitors were the better side before Randell's dismissal. Camiel Neghli shot narrowly wide and Casper De Norre headed over inside the first six minutes.
Ballard sent a shot flashing wide for City from Lorent Tolaj's 12th-minute knock down, but it was a rare moment of anxiety for Millwall goalkeeper Lukas Jensen.
Coburn claimed a penalty when going down as he rounded City goalkeeper Sam Tickle, but referee Lewis Smith saw no contact and booked the striker for simulation.
Randell's departure, having been cautioned for a foul 13 minutes earlier, saw Millwall crank up the pressure and Coburn's goal came as no surprise as the visitors went in front.
The Lions began the second half like the first and Metcalfe had a shot blocked in a crowded box before Coburn's close-range effort was saved by Tickle.
City had introduced midfielder Joe Williams for Hirakawa in an effort to tighten up, but looked wide open at the back as Coburn grabbed his second.
Moments later it should have been 3-0 as Arconte contrived to hit a post when unmarked following a cross from the right.
Tolaj shot over in a rare City raid, but their defence looked all over the place as Metcalfe headed over.
Skubala responded with a triple substitution, sending on Sam Bell, Emil Riis and Jed Wallace for Tolaj, Ballard and Sam Greenwood.
Wallace was soon in the action, firing inches over from a free-kick, but the damage had already been done and Millwall could have added a third but substitute Mark Syles sent an effort wide from 25 yards.
