Cosgrove, who has been linked with a move away from Aberdeen, picked up an injury just ahead of the season-opener on Saturday and set to be sidelined for several months.
Aberdeen have brought in Ryan Edmondson on loan from Leeds United, but could opt to go with Anderson for the visit from Steven Gerrard's men.
And Harper believes that the academy graduate is more than capable of leading his team's charge against a Rangers side who are looking to start off their make or break season with all three points.
"Cosgrove’s injury offers an opportunity for Bruce Anderson to step up and make his mark", Harper wrote in his column for the Evening Express.
"Anderson is definitely good and fast enough to lead the line against Rangers and he also scores goals.
"He has been scoring in the pre-season friendlies, so will be really fired up for the challenge of facing Rangers.
"Anderson will be confident in his own ability, so I would definitely consider starting him tomorrow."
Anderson featured in eleven league games for Aberdeen last season though he managed to find the back of the net only once.