The Danes came through late to edge the British team of Zac Purchase and Mark Hunter by a quarter-length and better the bronze they won in the Beijing Games in 2008.
The restart was caused when the clip that held Purchase's seat became twisted, causing him to slide. The race hadn't progressed more than 100 metres, allowing a restart to take place.
The French crew of Stany Delayre and Jeremie Azou held their hands up to signal a complaint when the incident happened but it was rejected by the race umpire.
A boat made an official protest after the race, according to world governing body FISA. That was also rejected, and FISA did not say which crew made the complaint.
"We are very good at dealing with things like that — that wouldn't have affected the outcome," said Hunter, who won titles at the last two world championships with Purchase. "These guys just rowed better than us today."
The British duo rowed to the side of the Dorney Lake course to get the seat fixed and then began quickest when the race restarted, establishing a lead of a second by the 1,500-meter mark.
But the experienced Rasmussen and Quist clawed their way back and overtook the British in the final 100 metres.
Rasmussen and Quist returned to form at the London Games after three lean years following their bronze in Beijing Games and world titles in the previous two years.
"We have been trying for many years now," Rasmussen said. "It feels amazing."
The silver for Britain was the host nation's ninth medal at the regatta — beating its previous best of eight at the 1908 Olympics in London.
New Zealand's Peter Taylor and Storm Uru took the bronze, more than three seconds behind the British.