England were 171 for four in their second innings, a lead of 404 runs, at tea on Saturday's third day.
Root was 97 not out and first innings centurion Ian Bell 16 not out.
The 22-year-old Root, playing his eighth Test, was preferred as England captain Alastair Cook's opening partner after making a maiden Test hundred as a middle-order batsman against New Zealand at his Headingley home ground in May, a match in which then opener Nick Compton struggled.
Their partnership meant that by lunch Australia, shot out for just 128 on Friday, already needed to surpass the Lord's Test record fourth innings victory total of 344 for one made by the West Indies against England in 1984, which featured Gordon Greenidge's unbeaten double century, if they were to pull off an astounding win.
England, 1-0 up in the five-match series after last week's 14-run victory at Trent Bridge, resumed Saturday on 31 for three, a lead of 264 runs.
After two days of blazing sunshine, conditions were overcast for the first time and offering a hint of swing.
Root, however, drove James Pattinson through the covers for four in a style reminiscent of former England captain Michael Vaughan, who played for the same Yorkshire club side as the Test novice.
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