South Africa and hosts Australia had been England's closest challengers at the halfway stage, but their performances in the 1000m range saw them slip down the standings to finish fourth and sixth respectively.
England finished with a score of 584-61v to retain the title they won in Glasgow four years ago, while Wales and Scotland picked up silver and bronze respectively.
The Welsh pairing of Chris Watson and Gareth Morris lifted themselves into the silver medal position during the 900m range, significantly improving on their seventh-place finish from four years ago. Scotland, meanwhile, win a second consecutive Commonwealth bronze medal.
England's Patel now has seven medals at the Commonwealth Games - including four golds - while Luckman has won his third gold.