Chasing 151, the UAE looked in trouble at 66-4 in the 13th over.
But Sharma, who made an unbeaten 74, and Soahib (51) put on 84 to help them achieve their target with two balls to spare in New Delhi.
Sharma played the anchor while Sohaib took on the opposition attack with four fours and four sixes to deny Canada and hand UAE their first win in two matches in Group D.
Sohaib raised his fifty off 28 balls but fell in the last over to leave Muhammad Arfan to hit the winning run to trigger joyous celebrations.
Pace bowler Junaid Siddique set up the win with figures of 5-35 in his four overs to limit Canada to 150-7 and was named player of the match.
"In the first innings, we thought they might be going for 160 or 170," said Siddique.
"We bowled very well in the last few overs and knew that 150 was a chaseable total."
Canada spinner Saad Bin Zafar impressed with 3-14 in his four overs to rattle the opposition top order.
But Sharma and Sohaib turned the chase on its head to make the UAE jump to third in the group ahead of winless Afghanistan.
New Zealand and South Africa head the group with two wins from two.
They clash in Ahmedabad on Saturday, where the winner will be the first team assured of a place in the Super Eights.
