The number of suspect couples referred to the interior ministry each year had more than doubled over the past three years to 2,145, a parliamentary committee said.
It urged that registrars, who officiate marriages, be given the power to refuse to marry couples they find suspicious.
"There is an industry of deceit in the UK which uses sham marriages to circumvent immigration control," said committee chair Keith Vaz.
"The estimated 10,000 sham marriages appears to be increasing at an alarming rate. One sham marriage can provide UK residence rights to an entire extended family who would otherwise have no right to be here."
Prime Minister David Cameron has vowed to cut immigration to the country ahead of a general election in 2015 in which migration is a touchstone issue.
A government estimate of 4,000 to 7,000 fake marriages a year was likely smaller than the true figure, independent chief inspector of borders and immigration John Vine told the committee.
Vine said the problem was "one of the biggest potential threats to immigration control".
Most commonly, people from outside the European Economic Area try to marry residents from within the zone to gain the right to live in Britain for themselves and their extended families.
Nationals from Lithuania, Poland, Portugal and Slovakia were reported to be those most commonly involved. Some Eastern European young women were offered cash and a new start in Britain for taking part.
The lawmakers called for more action such as asking embassies of the most commonly involved nationalities to issue warnings, and through well-publicised prosecutions.
"We are taking ever tougher action...to crack down on those who try to cheat our immigration system by abusing marriage laws," a Home Office spokesman said.