A method of laser welding a workpiece stack-up that includes two or more overlapping aluminum alloy workpieces is disclosed. The method involves controlling the power level of the laser beam during at least one of an initial stage or a final stage of advancing the laser beam along a weld path so as to limit a line energy of the laser beam during such stage or stages to being no greater than 10% above a line energy of the laser beam during an intermediate stage of laser beam advancement that is performed between the initial and final stages. By limiting the line energy during the initial and/or final stages of laser beam advancement along the weld path, excessive fusion of the workpiece stack-up assembly can be avoided in those locations to help protect against hot-cracking in the resultant laser weld joint.