The online firm was founded in 2011 and has more than 50 partner auction houses across six countries.
In the UK it has invested in Rushbrooks Auctioneers in Winchester and runs The Auction Hub.
Lawrences will continue to be run by chairman and managing director Helen Carless, with the current board of directors remaining in place.
Carless said: “I have been managing director at Lawrences for more than 30 years, since I lead a management buyout in 1992.
“Auctionet is a dream partner with whom to continue to develop the business and it is with great confidence I entrust Lawrences to the new owners.”
Carless added that Auctionet will build the team with plans to hire extra staff.
Niklas Söderholm, co-founder and CEO of Auctionet, said: “With this partnership we will strengthen our position in the UK market and help Lawrences implement an online strategy that is well proven across our group of European auction houses.”
The price paid has not been disclosed.