Click the "Advanced Search" or "More Options" button when looking for fares on an airline fare comparison website. These websites will show you the cheapest fares for any tickets.
Choose "First Class" for the class of travel when the website provides you with this search option. Because you are searching for the cheapest ticket in this class, the results page will sometimes lead you to Y-up, Q-up or Z, fares, which are all economy class fare codes. However these fare codes will automatically upgrade you to first class, usually at around the price of a full-fare economy ticket.
Check-in online at the earliest possible moment. If you didn't get a Y-up ticket, check to see if the airline is offering the option to buy an upgrade from your economy class ticket. The price of this upgrade, often between $50 and $100, will save you hundreds off the normal price of a first class ticket.
Check for the option to buy an upgrade at the airport upon arrival or at the gate if you were unable to buy it before the flight. By this point, if the aircraft has extra first class seats, the airline will be anxious to sell you an upgrade, no matter how cheaply.
Use frequent flyer miles to buy an upgrade into first class. If you have enough frequent flyer miles, it will not cost you a penny more than your original economy ticket.