Former Test batsman Marcus North defied the Bushrangers with a gutsy and classy seven-hour unbeaten 118 as WA finished the game 8-244, still 136 runs behind, to hold on for a draw.
Ahmed, who took 6-68 in the first innings, would have been looking for another haul on the last day to strengthen his chances of an Ashes Test debut.
But he went wicketless and didn't look dangerous.
In contrast, fellow leggie White, in superb form with the bat, bowled beautifully to claim four scalps in the last session to give Victoria a sniff of victory.
Shaun Marsh, who needed a big score to push his Ashes claims, made just four before becoming one of White's victims.
After impressive innings of 83 and 61 with the bat in this match, which followed being named player of the series in the state one-day tournament, White can lay claim to being the form player in the country.
It could perhaps push him into Test calculations, although he admitted himself earlier in the match that he didn't rate his chances.
White's only four previous Tests came in India in 2008, when he was used mainly as a bowler, despite having focused on his batting for most of his career.
WA looked set to comfortably hold on for a draw at 2-165 at tea, with North and first-gamer Will Bosisto (44) batting solidly.
But White, who bowled just one over before tea, almost proved a match-winner.
He trapped Bosisto lbw to break a 124-run stand.
After some brief resistance from Marsh (four), White then picked up his wicket, along with those of Sam Whiteman and Hilton Cartwright for ducks, in consecutive overs.
With his confidence soaring, he then took a sensational leaping left-handed catch at slip to help Peter Siddle dismiss Ashton Agar (six) with the new ball.
But Nathan Rimmington (10) and Jason Behrendorff gave North the support needed to deny the Bushrangers full points.