Bournemouth doubled their advantage when Marc Pugh lifted a cross into the area and the home side failed to clear their lines, leaving Grabban to turn in the loose ball.
It got better for Eddie Howe's men when Grabban was brought down by Darren Randolph in the area and they were awarded a penalty. The striker took the spot kick himself and notched his second of the game.
After the break, the visitors extended their lead further when Ian Harte got the fourth goal. He picked up the ball following a clearance and launched a 25-yard effort into the top corner.
Birmingham got a goal back when Macheda latched onto an Ibe cross into the area, took the ball down and fired it past Lee Camp from close range.
Macheda reduced the deficit again when he picked up the ball 25 yards from goal and drilled an effort past Camp, but it was too little too late for the hosts.