Beck struck a brace in Saturday's 3-1 victory at Yeovil to take his goal tally this season to three, following a crucial equaliser in the Capital One Cup victory against Ipswich in September.
The 18-year-old Beck is one of three teenagers to have impressed for the Cumbrians this term, alongside Brad Potts and Dave Symington.
He told the Carlisle News & Star: "I'm over the moon to get off the mark in the league and for the lads to get such a good away win.
"I was more pleased with the second one. I watched it come down, hit it first time and as soon as I hit it I knew it was going straight in. Luckily for me, that's what happened.
"I've been given my chance and hopefully I'll stay in the team and score more goals."