I did pretty much the exact same thing you're trying to do. Hope this helps!
If you want, you can replace the 555 circuit with an led. Works just the same.
Edit: Sorry about that. Kinda jumped the gun there. I guess I didn't have what would have helped, so I drew one. Use an SCR. Also, the value of R1 should be higher than the LDR when you have the laser shining on it, that way the voltage wants to go through the LDR rather than R1, and when the beam is broken, power goes through R1 and trips the SCR, which stays on until the power is cut, like with a switch. The first circuit I showed with the 555 was a light detecting circuit, but with a little switching around can work as a dark detector. Have fun!