Under the new rules, passengers who arrive from a destination not on the Government's travel corridors list will still need to enter self-isolation.
But they can reduce the 14-day period by paying for a test from a private firm after five days at a cost of £65-£120.
Results will normally be issued in 24 to 48 hours. This means people could be released from quarantine six days after arrival.
Mr Shapps said: "We have a plan in place to ensure that our route out of this pandemic is careful and balanced, allowing us to focus on what we can now do to bolster international travel while keeping the public safe.
"Our new testing strategy will allow us to travel more freely, see loved ones and drive international business. By giving people the choice to test on day five, we are also supporting the travel industry as it continues to rebuild out of the pandemic."
Tim Alderslade, chief executive of Airlines UK said the announcement provided "light at the end of the tunnel" for the aviation industry and those wanting to travel.
He predicted demand for air travel will "tentatively return" following the decision but said a pre-departure or domestic testing regime that can completely remove the need to self-isolate is "the only way we're going to comprehensively reopen the market".
Paul Charles, chief executive of travel consultancy The PC Agency, said: "It's a much-needed and long overdue step forward to helping the travel sector recover further.
"But we still have a complex jigsaw puzzle of restrictions around the world that need tourists to have a high IQ to understand. We need to see global consistency for travel to fully take off."