The 18-year-old Scot started the FA Cup last 16 game against Liverpool on 3 March and won plaudits for his calm passing and tenacious tracking during Chelsea's 2-0 win at Stamford Bridge.
On Sunday he silenced doubters that the FA Cup tie was a one-off performance with an assured dispaly during Chelsea's 4-0 waltz past a bedraggled Everton team.
"It's a bit surreal still, coming on and playing with top players," Gilmour told Chelsea TV.
"But this is something I've been dreaming of. And when I go out there I need to show that I'm not just a passenger. I need to be there and do something different to keep my position in the team."
Ambition
Chelsea's victory kept them fourth in the Premier League with nine games remaining of the season. They are three points ahead of Manchester United who beat Manchester City 2-0 at Old Trafford on Sunday.
Gilmour, who went to west London from the Scottish club Rangers in July 2017, said that established players at the club had helped him to settle in..
"That's something you always need as a young player, that confidence," Gilmour added. "If you don't have that confidence you're not going to play well, you're going to be a bit under pressure, be a bit naive.
"But I loved it and the first-team boys do really well with the youngsters."
Chelsea boss Frank Lampard has handed out a record eight debuts to academy products at Stamford Bridge this season.
Faith
Striker Armando Broja's late cameo off the bench against Everton meant Lampard surpassed the seven debuts awarded by manager Ted Drake in the 1956/57 season.
Tino Anjorin was another youngster to feature against Everton. The 18-year-old winger is among the contingent with Mason Mount, Reece James, Marc Guehi, Tariq Lamptey and Ian Maatsen.
Other academy graduates Tammy Abraham, Fikayo Tomori and Callum Hudson-Odoi have established themselves as first-team regulars.
"You can see that the manager will give the young players an opportunity and it's up to us to take it when we get the chance," Gilmour told the Chelsea website.
"It's exciting to be at Chelsea and I just want to take my opportunity when it comes."
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