Nathan Lyon was close to taking an early wicket when he caught Cook's edge from a delivery, but after the ball hit Haddin's pad, skipper Clarke was unable to leap forward in time to catch it.
The tourists were able to make the breakthrough before the close of play courtesy of Peter Siddle, whose delivery to Root hit the outside edge and went easily into the hands of Haddin.
Following Root's dismissal, England opted to bring in a nightwatchman in the form of Tim Bresnan, who was controversially dismissed from a Siddle delivery.
Bresnan (1) was given out following Australia's appeals, but he should have lodged a review as replays showed that the ball clipped the leg and not the bat.
Cook and Jonathan Trott (2) will return to the crease on day three.