Draper, 22, had to come through a serious examination but displayed great endurance as he outlasted the higher-ranked Kecmanovic to set up a quarter-final against American top seed Tommy Paul.
After Draper conceded a tight first set when he lost serve in the 12th game, the second continued into a tie-break in which the 55th-ranked Serb rallied from 5-2 down to reach match point.
But, on the brink of defeat when trailing 8-7 and then 9-8, world number 62 Draper clinched his fourth set point to force a decider.
Another recovery was required as that too went the distance, with Draper heading for defeat at 4-1 down in the concluding tie-break.
The Briton won five of the next six points to reach match point for the first time and he closed out an epic contest at the third opportunity to win a 5-7 7-6 (11-9) 7-6 (9-7) after three hours and 39 minutes.
However, British number two Evans was unable to join Draper in the last eight after losing 4-6 6-2 6-1 to Kazakh eighth seed Bublik.
Making his return from injury in Adelaide following the calf tear he sustained in Vienna in October, Evans, who impressed in his opening win on Tuesday, made a promising start to take the first set against the world number 31.
But the match unravelled rapidly for the 40th-ranked Evans after he fell a break of serve down in the second set, with Bublik cruising to victory by winning 10 of the last 11 games.
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