Lions winger David Martin broke down the left and Laird timed his run perfectly to stoop and head past Danny Coyne from six-yards after 71 minutes.
The result was harsh on Rovers who had the best chances on a stop-start afternoon full of petty fouls and yellow cards.
Rovers centre-back Ben Chorley forced David Forde to tip over his dipping volley after seven minutes and the Millwall keeper twice fisted clear from under his bar.
Clearly unhappy with his side's first-half showing, Lions boss Kenny Jackett hauled off the ineffective Tresor Kandol and Ali Fuseini for new loan signing Izale McLeod and Jimmy Abdou.
Charlton loanee McLeod instantly provided the cutting edge Millwall were lacking, latching onto Neil Harris' header and firing a shot against the upright on 50 minutes. But Tranmere continued to threaten, as diminutive forward Chris Shuker twice came close and Forde tipped over Edrissa Sonko's looping header.