The average rent in England and Wales in August surpassed the previous record high of £705 in July.
Arrears also increased, for the first time since April, with 10.7% of all rent unpaid or late by the end of August, up from 9% in July.
Unpaid rent totalled £300m in August, up 19.5% from £251m in July, but LSL says an August rise in arrears is a seasonal phenomenon.
On a monthly basis, rents increased fastest in Wales and the South-East, where they rose by 2.1%.
The next biggest increases were in London and the South-West, where they rose by 1.5% and 1.3% respectively, putting average rents in London at a new high of £1,025 per month.
Rents only declined in the Midlands compared to July, falling by 0.4% in both the West Midlands and the East Midlands.
The average yield across England and Wales was 5.2% in August, while the average total return was 2.6%.
David Brown, commercial director at LSL, says: “We are in the thick of the busiest time of year for the rental market, and red-hot demand for properties is driving rents up at their fastest monthly pace in the last 12 months.
“Recent graduates moving for their first jobs have further exaggerated the long-term and growing demand from frustrated buyers.
“With significant improvement in the number of buyers able to secure a mortgage unlikely in the foreseeable future, competition for rental accommodation will not drop and further rent rises remain on the cards.”
Brown says that new property investors are being drawn to the market by rental incomes and yields rather than thoughts of immediate capital gains.
He adds that while summer holiday spending can be used to account for the rise in arrears in August, rising rents and inflation are taking their toll on tenants.
Brown says: “While we expect arrears to fall back into line in the short term, the growth is indicative of the mounting pressure facing tenants. With rents rising so quickly, soaring inflation and an uncertain economic outlook, over the long term we anticipate that rental arrears will become a growing financial problem for landlords.”