Rumford's round included an eagle at the par-4 seventh hole, nine birdies -- one at the 18th and final hole -- and two bogeys.
The birdie at 18 vaulted him past Dyson entering Friday's second round.
"I got off to a shaky start in more ways than one," the Australian said, (but) today I just got my putter running."
Dyson started his bogey-free round with an eagle at the par-5 first hole, closed with a birdie at the par-4 18th, and in-between, made another eagle at the par-5 15th and three other birdies.
Miguel Angel Jimenez, Alexander Noren, Andres Romero, Christian Nilsson, Thongchai Jaidee and Danny Willett all shot 6-under 65 and are tied for third place.
Swim Week Miami Beach 2014 [PHOTOS]