The Hurricanes looked second favourites for victory in Hyderabad on Sunday when they required 44 from the final three overs chasing the Cobras' 184-6.
But Blizzard, alongside Jonathan Wells (17 not out), had other ideas as the pair smashed five of the next 12 balls for six to seal a remarkable victory with an over to spare.
The Cobras had been indebted to Richard Levi's 42 at the top of the order and some late hitting from Robin Peterson and Vernon Philander for their imposing total, but Ben Dunk made 54 at the start of the Hurricanes' reply before Blizzard and Wells saw their team over the line.
After winning the toss, Cobras lost Hashim Amla for eight to Ben Hilfenhaus (2-49), before Levi found support through Omphile Ramela (21).
Dane Vilas made 25 before also falling to Hilfenhaus, and it was left to Peterson (25 not out) and Philander (32 not out) to add 52 runs in the final four overs and propel the Cobras beyond the 180 mark.
The Hurricanes' opening pair of Dunk and Tim Paine put on 51 for the first wicket, of which the latter contributed just seven before being bowled by Charl Langeveldt (1-43).
Dunk and Blizzard added a further 40 until the opener was dismissed by Sybrand Engelbrecht, who returned impressive figures of 3-20 from his allocation of overs.
The Hurricanes then lost Shoaib Malik and Travis Birt cheaply and, with the chasing side struggling at 121-4 in the 15th over, the Cobras appeared to be well on top.
They reckoned without Blizzard, however, who thumped four fours and five sixes in his 48-ball knock to see the Hurricanes to their first win after going down to Kings XI Punjab in their Group B opener on Thursday.