Yeovil came up against a resolute Carlisle defence and the Cumbrians took a big step towards safety with a first-half header from Sean O'Hanlon and a second strike after the break by Beck, his sixth goal of the season.
It was a first win in seven games for Carlisle, which had a lot to thank central defensive pairing O'Hanlon and Danny Livesey for an uncompromising display at the back.
But it was 18-year-old striker Beck who came in for warm praise from Carlisle manager Greg Abbott.
Beck led the line due to the three-match suspension incurred by Lee Miller last week.
Abbott said: "Trotta and Wright-Phillips destroyed us last time we played Brentford, but Beck outshone both of them.
"No-one has probably heard about Mark Beck, but they will have heard about him after this.
"This is probably the best result since I became manager of this club.
"It's the proudest I've been. We have a group of players performing well above their station at the moment.
"We are coming out of a tough situation. No-one shirked anything and there was effort and determination."