Dragons 24-10 at BT Murrayfield thanks to tries from Tim Visser and the returning Dougie Fife, while Tom Heathcote also kicked 14 points.
The win sees Edinburgh climb to eighth in the table and 24-year-old Gilchrist, who has made 56 PRO12 appearances for the club since joining in 2011, has called on his team-mates to up the ante as they head into the European break.
"I really care about this club and there are a lot of other guys who do as well," Gilchrist told The Scotsman.
"We always want to win and it's on us. Alan Solomons has identified the five guys that he sees as being leaders and he wants us to drive the team's standards on.
"The coaching staff are excellent here and we're always well prepared for every game.
"We do our homework, we train well and now it's about the execution on a Saturday. We need to be playing with a real energy and that comes from the senior guys."
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