The property website found interest in homes for sale is now down just seven per cent compared to the same point last year.
Last week, the government said people could again move homes and view properties after the market was put on standstill in March.
Almost 11,000 new sales listings have been put on the market since the first day of reopening, Rightmove said.
This is 65 per cent down on the same week last year, but a 90 per cent increase compared to the weeks when the market was in lockdown.
Total available stock for sale is down 2.5 per cent since before lockdown.
Only one per cent of properties up for sale have been reduced in price over the past week, compared to two per cent over the same week last year.
Average asking prices of all property up for sale are up two per cent on last year, the same as before the lockdown. The current average UK value stands at £311,828.
The biggest increase in buyer searches compared to last year are in Manchester, Liverpool and Leeds.
Record-breaking demand for rentals
At the same time demand for rental property is up by 33 per cent compared to this time last year.
Monday 18 May had the highest level of rental demand ever recorded in one day on Rightmove
And total available stock for rent is now up 13 per cent since before lockdown.
Asking rents are currently up 2.1 per cent on 2019, down from an annual rate of three per cent before lockdown. They are on average £1,089 per calendar month.
Rightmove’s housing market expert, Miles Shipside, said: “One week on from the surprise opening of the housing market and agents are still showing caution and quite rightly putting safety first, but many had already started to prepare during lockdown and so have been able to start reopening their branches and conducting viewings.
“The activity jump on Rightmove proves the determination of home-movers to continue with their plans over the next few months.
“The pause button put many of the usual metrics we report on out of kilter and so a week’s worth of data should only be taken as an early indication of activity, but it’s certainly encouraging.”