In a perfectly competitive market, a firm that invents a new method of production that lowers its marginal costs will either have lighter losses if it is unprofitable or higher profits if it is profitable.
What happens to the price it charges for its widgets is entirely up to the firm's owners or management. Under perfect competition, their temporary marginal cost advantage will dissipate quickly, as this condition assumes no lasting information asymmetries.
Lower prices, all things being equal, should allow them to sell more units and maximize short term profits so they may lower their price to try to increase market share. Perhaps they are already competitive or dominant in the market in which case they may leave their price the same and enjoy higher profits for a time.
No matter what they do with the price under perfect competition the innovation in productivity will soon spread to competitors, however, and the marginal cost advantage it provided will vanish. Remember under perfect competition all profits go to zero, and all firms eventually must exit. That is one way we know imperfect competition and imperfect information are ubiquitous marketplace phenomena.