Jump into the meat of the situation with your first line. A character saying, "It's one year today," for example, pulls us right into the moment, and as an audience we're wondering --- one year since what? This will hold us until the next beat. You want to hook your audience and keep pulling them along on the journey with every line.
Set up conflict and character quickly. Some sort of discord between characters expressed in the first few lines and in the action heightens our interest and starts to fill in a little of these people. Who are they? we wonder. And most important, what do they want?
Raise the stakes. Your characters may be discussing that it's one year since the windows were washed. We probably won't care very much once we find that out --- unless the foggy windows are meant as some sort of protection. Then we're intrigued. At least until the next beat. Keep the stakes high; this has to be an extraordinary moment, not an ordinary moment.
Take action. Your characters and conflict are quickly established and the stakes are high. But if the situation stays status quo for the rest of the one-page drama, it will fizzle, no matter how witty the dialogue. Characters need to be in pursuit, taking action, making choices, even if the action is internal, such as making a decision. Action not only moves the play forward, it helps us understand who these characters are. We get to know them by the actions they take, not necessarily by what they say. In a one-page play, you can't build to an action. Just take it.
Add depth. In Chekhov's plays, you hear music from another room; you hear the crowds outside and mention of the fires taking place there. You see and feel more than what is before you. This can be helpful in a one-page play to expand the limited scope. The mention of something blooming in the garden, for example, sets the time of year, perhaps even the month or locale, and gives us a mental image beyond what we see.
Bring things to a head. This is the moment we have been waiting for. Will the main character succeed in getting what she wants? In a one-page play, the climax has to be accomplished in a just a few lines with a strong definite action. Remember, an action doesn't have to be physical; it can be a decision. Try to avoid outcomes that are predictable. That's the least interesting to an audience. At the same time, magical outcomes also tend to be disappointing.
Cut the extraneous. Allow yourself to overwrite in your first draft, even if you are exceeding the one-page length. Just get the story out. Then go back over your script and eliminate everything that isn't needed. Remember that every action should connect to a previous one and to the next one: cause and effect. As David Ball says in his book "Backwards & Forwards: A Technical Manual for Reading Plays," you should be able to read a good script backwards, and for every action, find the preceding cause of that action. If it doesn't connect, you are diminishing your focus.
Allow for the unexpected. There's no real formula for writing a play, one-page or 100 pages. You can map it out and make sure it contains all the elements of drama as outlined by Aristotle back in 350 B.C., but always allow for the unexpected --- as long as it makes sense based on what has come before. If it surprises you, it probably will surprise your audience. And there's nothing like surprise and wonder. Break a leg!