The £149 billion total of sales recorded represents nearly double the value of homes sold subject to contract in the same period in 2020 and 2019, Zoopla said.
And many homes are selling for above the asking price as strong activity is eroding the supply of homes available.
It said the total number of homes on the market and not yet under offer was around 30 per cent down in early April compared with average levels during more “normal” times between 2017 and 2019.
Three and four bedroom houses have recorded a particularly sharp fall in supply.
In the North West, there was a whopping 42 per cent drop for four-bedroom homes year on year.
This was also down by 58 per cent in Scotland, and in England by 44 per cent in the South West, and 40 per cent in the South East, according to the research.
There is a risk the lack of family houses for sale will limit sales growth in the second half of 2021, Zoopla added.
Glasgow, Bristol, Nottingham, Stoke and Middlesbrough are particularly busy sales markets generally, it said.
Meanwhile, Manchester, Liverpool, Leeds, Nottingham and Leicester are registering particularly strong house price growth of more than five per cent year-on-year.
A new five per cent deposit mortgage scheme, backed by the Government, was launched last week to give first-time buyers a boost.
Grainne Gilmore, head of research at Zoopla said: “More flexible working arrangements open up new opportunities for home owners to move to a further-flung location.
“At the same time, the rollout of the 95 per cent mortgage guarantee will mean more demand from first-time buyers, fuelling demand without replenishing supply.”
Meanwhile, latest figures show one in six homes were snapped up in March for more money than the original asking price.
It marked the highest proportion of homes selling for above the asking price in about seven years.
One senior Lancashire estate agency professional said the trend was being reflected in many areas.
She said: “Houses are being snapped up very quickly – and often end up in a bidding war, going for much more than the vendor thought was possible.”
The number of sales agreed was also the highest for the month of March since 2007, NAEA (National Association of Estate Agents) Propertymark said.
An average of 12 sales were agreed per estate agency branch – the highest figure for March since 2007 when the number was also 12.
More than a quarter of sales in March were made to first-time buyers – the highest figure since June 2020.
A stamp duty holiday, which was due to end on March 31 2021, was extended in the recent Budget.
NAEA Propertymark said the 16 per cent of properties selling for over the asking price in March was the highest figure since May 2014, when 19 per cent of properties sold for more than the original asking price.
A third of homes sold for less than the original asking price in March – the lowest number since records started.
In February 2021, just over one in 10 of properties had been selling for over the asking price.
Mark Hayward, chief policy adviser at Propertymark, said: “The imbalance of supply and demand means it’s an extremely strong sellers’ market.
“Properties are selling quickly and for over the asking price.”