Former Foxes skipper Ackerman was the game's top scorer with 66
Yorkshire made heavy weather of it at Headingley as they got their Twenty20 campaign off to a winning start.
HD Ackerman was Leicestershire's top scorer, batting through for an unbeaten 66, from 56 balls, out of their 148-3.
Yorkshire then looked well on course at 110-2, only to lose five wickets for 15 runs, including Michael Vaughan for 35.
But young Ajmal Shahzad, who hit 16 off the penultimate over, and keeper Simon Guy saw their side home on 149-7 to win with just one ball to spare.
Needing 22 off the last two, man of the match Shahzad had a close, match-turning escape.
After taking two off Jim Allenby's first delivery, he drove his next ball high to long on where James Taylor managed to get both hands to the ball, only to see it slip from his grasp and go over the rope for six.
To the delight of the 5,300 crowd, Shahzad then celebrated with a sizzling straight drive for four.
And, with 16 having come off the over, the home side now needed just to six to win off the final six balls.
Guy failed to score off White's opening ball but he then edged his next to the third man boundary. And after taking a single, it was left to Shahzad to dash the winning run with one ball to spare.
Yorkshire now turn their attention to Tuesday evening's Roses battle with Lancashire at Old Trafford (1730 BST).
Yorkshire batsman Jacques Rudolph told BBC Radio Leeds:
"This is what Twenty20 brings to the party.
"We lost five wickets in a short space of time but we pulled it through.
"Now we've got to hold it together against Lancashire tomorrow.
"They're a quality team but so are we, so it should be quite competitive."
Leicestershire head coach Tim Boon: "We just missed that winning edge.
"They restricted us and we were probably 15 or 20 runs short.
"The guys are down but there were some really good things with some good young players who will play for a number of years.
"We lost the game but I still feel we are in a good space."
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