If firms minimize costs in economics, then does that mean they must invariably deliver low-quality products?

The statement given above is not valid. In economics, the firms will minimize the cost of good quality product to keep the demand for their goods high or to prevent the competition faced by the other firms in the market. The firms will only reduce the price of their goods only if they have a low cost of production. The new consumers in the market will too respond to the low prices and this will help the firm by increasing their sales and capturing a higher market share. If the low price is charged for the poor quality product, then the consumers can easily identify the gap between the prices.