It was Scotland's third successive Test win following victories over USA and Canada respectively, and Gilchrist, who captained the team, could not contain his pride.
"It wasn't perfect but we showed a bit of character in the last 10 minutes and that is something I am really proud of," the second-row is quoted as saying by Sky Sports News.
"I was just saying 'keep believing in what we have been doing all week'. I think it was a scrum that was the turning point, we pushed them off their ball and got three points to put us within seven and then great individual play and a great score took us right back in it.
"Then we had to score again and our forwards really stepped up. I am really proud."
Scotland head to South Africa next week as they search for a fourth consecutive Test win.