Southwell has not recovered after a collision with Lee Byrne in the defeat to Wales while Ansbro received a concussion while on club duty for Northampton Saints.
Lamont - who could have replaced Southwell at full-back - was also injured while playing for Toulon in the Top 14.
Even though Robinson will be working with limited resources against Ireland, he called on his team to rediscover their winning touch against Ireland.
He said: "We have made the decision now that all three of our injured backs will miss this weekend's international so that brings clarity to our preparations.
"We wish all of them a speedy recovery.
"Everybody was hurting after the display against Wales but we must now focus on the challenge that Ireland bring and there's absolute determination to bounce back against them at Murrayfield on Sunday."
Robinson's problems don't end there with six forwards requiring further medical assessment including Jon Welsh, Robert Harley, Ross Rennie, Nathan Hines, Richie Vernon and Dougie Hall.