"We're delighted to sign him," Strachan told the club website.
"We watched him on a few occasions and always liked what we saw. He's a good ball player and he's got a good left foot. He's a good player."
Kink has been a regular fixture in the Estonia squad after making his international debut against Northern Ireland in 2004.
At club level Kink has European experience with Gyori as well as Russian side Spartak Moscow and domestic outfit Levadia Tallinn.
Comfortable on the left of midfield, Kink will provide competition for fellow signing Andrew Halliday, and is set to link up with the Teessiders on their return from a mini-tour of Germany next week.
Source: BBC Sport
Source: BBC Sport