Leroy Lita scored twice, with Dave Kitson, Gary O'Neil and Mark Yeates also finding the net for Boro in a resounding triumph.
They went ahead just past the half-hour mark mark courtesy of Kitson's third goal in two matches.
The on-loan Stoke City striker lost his marker and steered O'Neil's free-kick into the corner of the net.
It was a deserved breakthrough for Strachan's side, who had almost gone in front eight minutes earlier when Yeates' 25-yard drive struck the underside of the bar.
The visitors also went close when O'Neil shot wide and Lita fired straight at keeper Radek Cerny after being put through by Kitson, so it came as no surprise when Boro eventually opened the scoring.
And they went two-up just minutes into the second half. David Wheater's header from O'Neil's left-wing corner was handled by Ben Watson, and Lita stepped up to smash home the resulting penalty.
That prompted Rangers manager Jim Magilton to make a double substitution, sending on Rowan Vine and Patrick Agyemang.
Within a minute, Vine found space on the left and crossed for Agyemang to score from close range.
But any prospect of a comeback was crushed when Mikele Leigertwood gave the ball away, and Lita cut in from the left to fire in his second goal.
Worse followed for the London side when Lita shot against the post and O'Neil tucked away the rebound.
And to rub salt into their wounds, Yeates hit the net with a late 25-yarder to complete the rout.