Warner's exit for 52 left Australia 97 for two at the interval at Cardiff's Sophia Gardens and needing a further 315 runs to reach what would be a new Ashes record fourth-innings winning total of 412.
Steven Smith, the world's number one ranked Test batsman, was 29 not out after off-spinner Ali had Warner lbw as he shuffled across his stumps to a ball that came in on the angle rather than turned away from the left-hander.
The most any side have made to win in the fourth innings of an Ashes Test was Australia's 404 for three at Headingley back in 1948 when Arthur Morris scored 182 and Donald Bradman, widely regarded as cricket's greatest batsman, an unbeaten 173.
But Ashes-holders Australia, who dismissed England on the stroke of Friday's stumps for 289 in their second innings, knew they had two days in which to rewrite the record books as their openers walked out to bat under blue skies on Saturday.
Fortunately for England, it was not an expensive miss with Rogers out for 10 when he edged a sharply rising delivery from Stuart Broad low to Ian Bell at second slip to leave Australia 19 for one.
It was a fine reward for Broad during a miserly opening spell of one for 10 in six overs.
England captain Alastair Cook brought Ali on as first change but when he dropped short, Warner pulled him for six and cut him for four off consecutive balls only for the spinner to have his revenge on his return.
