Gillette, which dominates the global razor business, has long followed a simple and lucrative strategy: Add new features and raise prices.

But the 115-year-old brand is changing tactics this month by slashing prices and putting a new focus on its cheaper products. The Procter & Gamble Co. unit hopes to stop defections of its U.S. customers to online startups like Dollar Shave Club and Harry’s that sell lower-priced razors and blades. Gillette’s plan to cut prices by as much as 20% jolted Wall Street. “An act of desperation...