Alexander took 4 for 27 as the Dolphins successfully defended a score of 145 for eight, with their seven-run victory narrowing the gap to the table-topping Cobras to just six points.
With the Dolphins having two games in hand, their hopes of overtaking the Cobras to earn a home final have become much more feasible.
The visitors won the toss and elected to bat first. They duly set the platform for a strong total as they reached the halfway mark of their innings on 82 for one.
Cameron Delport maintained his excellent form by hitting 49 from 37 balls, but had his innings ended by Justin Ontong, whose first bowling stint of the tournament yielded figures of 2 for 9 from two overs.
The Cobras bowlers backed their captain up as Sunil Narine dismissed danger man David Miller on his way to figures of 3 for 28, and Beuran Hendricks picked up his 20th and 21st wickets of the competition.
With the Dolphins managing just 63 runs in their last 10 overs they appeared to have fallen short of a truly competitive total, but Robbie Frylinck dismissed Richard Levi cheaply and Alexander got the better of Stiaan van Zyl and Ontong in successive overs.
Khaya Zondo and Daryn Smit also chipped in as the Cobras fell to 45 for five, which became 46 for six when Alexander had Hashim Amla caught for 26.
Justin Kemp attempted to bring the Cobras back into the game on his return to the side, but his 56 not out from 33 deliveries was not enough to rescue the innings as the hosts finished on 138 for eight.