The right-arm fast bowler, who completed a 10th first-class, five-wicket haul, claimed three of the remaining four wickets required skittling the second-placed hosts for just 102 in pursuit of a mammoth 501 victory target.
Only 45 minutes were required by the Durbanites after the hosts resumed on 73 for six.
Alexander trapped Dieter Klein leg before wicket with the ninth ball of the day without adding to his overnight 17. He then accounting for Junior Dala (5) and Eddie Leie (4), both caught by David Miller.
Number 11 Pumelela Matshikwe was last man out when he was cleaned up by Kyle Abbott for the Dolphins to seal an emphatic win.
Meanwhile in Port Elizabeth, Wayne Parnell and Simon Harmer shared nine wickets as the Warriors also secured a maiden win this season, dishing out a 225-run thrashing to the Titans.
Proteas’ quick Parnell finished with 5/48, the third five-wicket haul of his career, while Harmer took 4/57 to leave the Pretoria side all out for 214 in pursuit of 440.
The visitors had started out the day on 84 for four, but were immediately under pressure as two wickets fell without a run being added.
Albie Morkel was out for his overnight 29 Parnell drawing an edge to Harmer at second slip before Mangaliso Mosehle edged the left-arm paceman to Jon-Jon Smuts at second slip for a duck.
The other overnight man Francois le Clus (58) then managed to add 41 with top-scorer Roelof van der Merwe for the seventh wicket, before Ayabulela Gqamane dismissed the former, caught by Ryan Bailey at mid-off shortly after completing his 18th first-class 50.
Harmer then stepped up to mop up the tail with the off-break bowler taking out Shaun von Berg (15), Rowan Richards (18) and Ethy Mbhalati (13). Van der Merwe remaining unbeaten on 61 as the game ended shortly before lunch.