The home team recovered from 8-0 behind to secure the Six Nations victory, with Hamilton branding the scrum "absolutely immense".
"We have put a lot of work into our line-out and scrum. I thought our scrum was absolutely immense. I think a special mention has to go to Geoff Cross. He has come in and done a magnificent job. Credit to all the forwards," Hamilton told reporters following the match.
"We really dug deep today and I think you can see that win means a lot to us. We knew Ireland notoriously start well, so we hung in there in the first half.
"To go in at half-time with the score at 3-0 to Ireland was probably a good outcome for us. We went out in the second half and had to dig deep, and we took our chances."
The result left Scotland in third position, two points behind leaders England.