Strachan sealed a maiden win at the fifth attempt as Leroy Lita’s brace and goals from Dave Kitson, Gary O’Neil and Mark Yeates sank Rangers.
Kitson and Lita put Boro 2-0 up before Patrick Agyemang pulled one back. But instead of folding under the pressure, Boro stepped up a gear.
Yeats had hit the bar before Kitson (30) finished a free-kick from O’Neil. And Boro made it 2-0 when Ben Watson handled in the area and Lita (48) scored from the spot.
QPR sub Rowan Vine then crossed for fellow sub Agyemang who swept the ball home.
But Lita (60) rifled home an unstoppable shot and he was denied a hat-trick when another drive crashed off the post but O’Neil scored from the rebound.
And Yeates’ curler made it 5-1.