The 42-year-old, who will leave his role with Leicester Tigers, has signed a three-year contract with the club that will officially begin at the start of July.
Dawson told the club's official website: "We're delighted that Matt has agreed to become head coach and we believe that he will be carry on the fine work which has been done over the course of the last eight years.
"Matt joins Leinster with an impressive pedigree under his belt at club level with the Brumbies - who were provincial champions in 2006 - and Leicester Tigers - who have boasted the league's best defensive record (in 2009-10) and most tries scored (between 2008-13) - during his time on their coaching team.
"We were impressed by Matt's vision for Leinster Rugby and speaking to him it is clear that he is excited about the opportunity. We look forward to working with him over the coming years."
O'Connor replaces new Ireland coach Joe Schmidt at the helm.