Morris's was the more spectacular innings as he plundered three sixes in making 66 off 46 balls. It was only the sixth time he had passed 50 in 51 Sunday innings for Durham, and it carried his side to 146 for six, which proved enough to secure a two-run victory.
While Dawson was scampering to 50 off 40 balls, Gloucestershire were marginally ahead of the required rate and his opening stand of 72 with Nick Trainor was the perfect launching pad.
But when both fell to Mike Foster, returning after injury, the visitors lost their way until they took 17 off the penultimate over, bowled by John Wood. That left 14 needed, but Mark Alleyne was run out, and they finished on 144 for eight.