Derby's Will Hughes produced a vital block on 14 minutes to deny Lions defender Shane Lowry, who let fly after picking up Liam Feeney's corner.
Kenny Jackett's men kept up the early pressure, forcing another save from Legzdins through New Zealand international Chris Wood three minutes later.
Millwall dominated the attacking play throughout the first half and took a deserved lead before the break through Feeney.
Chris Taylor crossed the ball in from the left wing and midfielder found the back of the net. Feeney appeared to be offside, but the goal stood despite Derby's protests.