Shoppers who bought the lasagne products, which are made by French food supplier Comigel on behalf of Findus, have been warned not to eat them.
Findus tested 18 of its beef lasagne products and found 11 meals containing 60% to 100% horse meat, the FSA said.
Retail giant Tesco and discount chain Aldi withdrew a range of ready meals produced by Comigel over fears that they contained contaminated meat.
The FSA said there is no evidence to suggest the horse meat found is a food safety risk.
But the agency confirmed tests have been ordered on the lasagne for the veterinary drug phenylbutazone. Animals treated with "bute" are not allowed to enter the food chain.
An FSA statement said: "As part of its ongoing investigation into mislabelled meat, the Food Standards Agency has confirmed that the meat content of beef lasagne products recalled by Findus has tested positive for more than 60% horse meat.
"Findus withdrew the beef lasagne products after its French supplier, Comigel, raised concerns about the type of meat used in the lasagne.
"We have no evidence to suggest that this is a food safety risk. However, the FSA has ordered Findus to test the lasagne for the veterinary drug phenylbutazone, or 'bute'. Animals treated with phenylbutazone are not allowed to enter the food chain as it may pose a risk to human health."
Anyone who bought the lasagne products should return them to the shop they were bought from, the FSA said.
Findus UK withdrew its 320g, 360g and 500g lasagne meals from supermarket shelves as a precautionary measure earlier this week.
It came after Comigel alerted Findus and Aldi that their products "do not conform to specification".
It advised them to remove Findus beef lasagne and Aldi's Today's Special frozen beef lasagne and Today's Special frozen spaghetti bolognese.
Tesco also decided to withdraw its Everyday Value spaghetti bolognese, which is produced at the same Comigel site.
Findus UK apologised to customers tonight and said anyone who bought the affected lasagne products could get a full refund.
A spokesman said: "We understand this it is a very sensitive subject for consumers and we would like to reassure you we have reacted immediately. We do not believe this to be a food safety issue.
"We are confident that we have fully resolved this supply chain issue. Fully compliant beef lasagne will be in stores again soon.
"Consumers who have purchased the product should call our Findus UK Customer Care Line on UK 0800 132584, ROI 1800 800500 or email email@example.com for a full refund.
"We would like to take this opportunity to apologise to our customers for any inconvenience caused."
The latest development in the contamination crisis comes days after supermarket chain Asda withdrew products supplied by a Northern Ireland company which was storing meat found to contain a high proportion of horse DNA.
Newry-based Freeza Meats had been storing the consignment of meat, which was labelled as beef, on behalf of a supplier in the Irish Republic - Co Monaghan-based meat trader McAdam Foods.
Two tested samples were found to contain 80% horse meat.
McAdam Foods has insisted it had no knowledge that any of its meat contained horse DNA. It claimed the contaminated produce originated in Poland.
The meat had not entered the food chain and was not destined for Asda stores.
Asda acknowledged that no trace of equine DNA had been found in products made by Freeza Meats, but said it was still temporarily removing its burger range from its stores as a precaution.
The Irish meat-processing industry has been rocked by the horse meat crisis, with a number of suppliers being caught up in the scare.
Authorities on both sides of the border have pledged to restore the sector's battered image, while police in the Irish Republic have launched an investigation.
Irish Cattle and Sheep Farmers' Association beef committee chair Edmond Phelan expressed his concern at the situation.
The FSA added: "People who have bought any Findus beef lasagne products are advised not to eat them and return them to the shop they bought them from."