Cochrane, Max Bird and Jay Stansfield netted with left-footed finishes before half-time to ease the loss of seven-goal striker August Priske with a knee problem.

It put City on their way to their sixth win in eight and extended their unbeaten Championship home start to seven games.

Cochrane netted when he lashed home a loose ball from 30 yards when Millwall failed to clear a long pass and Phil Neumann’s shot hit a defender. Goalkeeper Max Crocombe was motionless on his line as Cochrane netted for the second successive game.

Millwall forced City goalkeeper Ryan Allsop into a flying save from Caleb Taylor’s glancing header following a long throw and Kyrell Lisbie hooked over from the rebound.

City made the visitors pay by doubling their lead in the 16th minute. Bird steered home a left-footed drive from 15 yards after sidestepping Derek Mazou-Sacko before taking on Taylor to celebrate his first start for Birmingham.

Blues continued to threaten and Stansfield’s curling shot was palmed away by Crocombe before Jhon Solis’ downward header bounced straight at the Millwall goalkeeper.

But in between Millwall wasted the opportunity to reduce the arrears when former City striker Lyndon Dykes flashed a rising drive over the bar as he fell. Jake Cooper then scooped a similar effort over from Lisbie’s low cross.

Blues made them pay by making it 3-0 with a fine breakaway goal in the 29th minute. Stansfield took Bird’s pass and ran at the defence before cutting inside Cooper and sending a left-footed drive through Taylor’s legs into the bottom corner from just outside the box.

The home side continued to hold the upper hand following the restart, although it was Millwall who showed first. Cooper fired straight at Allsop but Crocombe remained the busier of the two keepers.

Crocombe had to dive to his right to keep out Stansfield’s shot on the turn from the edge of the box and then he smothered another Stansfield effort and Patrick Roberts’s curling shot.

Millwall’s closest effort was a thunderous shot from Camiel Neghli that shook the crossbar in the 75th minute.

Substitute Carlos Vicente went within a whisker of making it 4-0 when his angled shot came off the inside of a post and rolled back to Crocombe.