The hosts enjoyed almost complete dominance in the first half, with Ibrahim Osman at the heart of their good play.
The winger grazed the crossbar with a cross-cum-shot after just two minutes before teeing up Roberts to fire home from close range three minutes later after he nutmegged his opposite number and sent a low cross into the box.
Only some acrobatics from Amadou Mbengue stopped Blues from doubling their lead shortly after when defender Christoph Klarer connected with a free-kick at the far post and hooked the ball towards goal.
R’s keeper Joe Walsh had to be alert to deny Osman, Roberts and then Jay Stansfield as the one-way traffic continued, with Ethan Laird also seeing a header come back off the post.
QPR had gone three games without scoring before kick-off and failed to register a single effort on goal until the 71st minute.
Blues continued to dominate after the break and Marvin Ducksch might have done better when Stansfield’s cutback found him unmarked on the edge of the box.
Harvey Vale curled wide as QPR finally threatened the Birmingham goal with less than 20 minutes left on the clock.
Substitute Kwame Poku added some life to the visitors’ attack and flashed an effort across goal in a rare moment of pressure, but the victory for Chris Davies’ side was more comfortable than the scoreline might suggest.
