But Morkel and Elgar combined in a 195-run partnership - a record fifth-wicket stand both for the Titans and the ground - that put the visitors within sight of victory.
Although Elgar fell for his second 100 (119 balls, 3x4s, 3x6s) in as many innings, Morkel responded with another volley of boundaries that saw the Titans home with 17 balls to spare.
Morkel’s unbeaten 134 (103 balls, 8x4s, 7x6s) was his first List A hundred, adding to the 15 fifties he has scored in 193 games in the format. He was particularly powerful when hitting over the offside off the back foot, and his strike rate ensured that the asking rate never rose above 7.2 to the over.
The Cobras had shared or won the last three one-day trophies, and will reflect on their inability to cash in on the excellent platform built by Richard Levi and Andrew Puttick.
The pair put on 180 for the first wicket in 31.3 overs after the Cobras had won the toss and elected to bat - a record for both the Cobras and the ground - but the innings disintegrated after Levi was trapped lbw for 104 (113 balls, 11x4s, 1x6) by Tabraiz Shamsi.
Three batsmen were run out, including Puttick for 69, and Dane Vilas (40 from 25 balls) was the only batsman able to inject some momentum back into the innings as the hosts added just 104 runs in the last 112 deliveries for the loss of eight wickets.
The Cobras still would have expected to have enough runs on the board, particularly after Rory Kleinveldt removed Henry Davids with the first ball of the innings and also picked up Jacques Rudolph in a threatening opening spell, but Elgar and Morkel had other ideas.