Wicketkeeper Ramon Sealy was no nearly their hero, as he made a terrific 81-ball 84 which included eight fours and two sixes, and took them to within 39 runs of their target of 253 before he was fifth man out.
But Norway's skipper Raza Iqbal played an even more decisive role, also making 84 and then effectively winning the match with the ball. His second five-over spell engineered a collapse which took the Caymans from 196 for three with nine overs remaining to 244 for nine at the close, giving the Norwegians an 8-run victory.
17 runs were needed from his final over, and with eight coming from the first four deliveries he was able to finish off with two wickets and end with five for 48, appropriately taking the Man of the Match award into the bargain.
It had looked so promising for the Caymans going into the last ten overs, with solid contributions from opener Sacha de Alwis (40) and the rest of the top order, but in the end Iqbal and his side were able to join Vanuatu in the semi-finals and keep alive their hopes of progressing to Division 5.
Electing to bat first, Raza Iqbal shared a 58-run opening stand with Pratheesh Thangavadivel (31), and then he and Suhail Iftikhar added 122 for the second before the Norwegian keeper fell to Ryan Bovell for 63, including seven fours and two sixes.
The captain was the fourth to go, departing with the total on 196 and more than 13 overs left, but of the remaining Norwegian batsmen only Vithyeswara Sarma (20) and Damon Crawford (20 not out) were able to reach double figures, and they were unable to produce the big finish which would have put the game beyond the Caymans' reach.
Still, 252 was a reasonably challenging target, and in the end it proved to be just enough. Norway will now meet the all-conquering Guernsey side at Frinton-on-Sea on Friday for a place in Saturday's final.