Note: This achievement's description is outdated. The achievement now gives you three minutes to escape; not two.
Spawn in as a Scientist. Search for the green Zone Manager card. It is not impossible to do this as a D-Class or Guard, but with how many variables there are as-is, it is significantly more difficult to get this achievement as anything other than a Scientist.
You can find a guaranteed Zone Manager card on a table in SCP-012's Containment Chamber. Don't drop your Scientist card; you're going to need it later on.
Do not go to upgrade your card. Instead, search for either of the exits. When you get up to Heavy Containment, immediately search for SCP-096's Containment Chamber. It is a small room, and the door resembles a default Heavy Containment door. Use your Scientist card to open it, and inside there will be a blue Luitenant keycard sitting on the ground.
Continue on and search for the Entrance Zone. Gate B is preferable, as it is closer to the escape area. Gate A is not impossible, but as you are relying on RNG and a randomized map, finding Gate B may save you much-needed time. Use the blue Luitenant card to open the gate. (The elevator will already be called down at Gate B, saving you even more time).
-Note: This is one of the more difficult achievements in the game as it is purely based on RNG, luck, and pre-learned knowledge of the game's escape routes. I gave up trying to get this achievement, turning to cheat by spawning myself at the exit using mod commands. Good luck getting this, but it will probably take many hours to do so.