The 28-year-old will officially join on June 1 when his contract with the Dingwall club expires.
Stuart McCall moved to sign the former Celtic player after his impressive performances helped County to fifth place in the Clydesdale Bank Premier League.
Lawson, who had loan spells with St Johnstone and St Mirren before joining County in 2007, told MFC TV: "It's great to have my future sorted out. There is a lot of uncertainty around Scottish football so it's really pleasing to have things sorted early doors.
"I enjoyed my time at Ross County. I have been pleased with my performances and thankfully the manager spotted that and came in to sign me.
"He is probably the main reason why I joined. I met him and he was keen to bring me here which was great and obviously the boys have enjoyed a fantastic season and I am keen to try and continue that next year."
Motherwell have also confirmed their full squad situation with new deals offered to the likes of James McFadden and Fraser Kerr, who was on loan from Birmingham.
Definitely leaving the club are Birmingham-bound Darren Randolph and Nicky Law, both of whom rejected offers, while Steven Saunders, Omar Daley, Steven Hetherington, Josh Watt and Nicky Devlin were released.
McCall is awaiting answers from McFadden, Kerr, Tom Hateley, Chris Humphrey and Michael Higdon, with defender Simon Ramsden signed up last week.
Kallum Higginbotham will return to Huddersfield following his loan spell while under-20s players Adam Asghar and Dominic Green have been freed.