Don't compete on price, per se. Compete on value. Value is related to price, but it's not the same thing. Identify the customer segment(s) you believe you're building for, and that your competitors might not be -- either because they have to go broad, or because they're focused on a segment or two themselves. Then really nail the value prop and solution for the segments you're targeting. You'll probably want to price cheaper than the big guys, but it's not to position yourself as "basically the same thing, but less expensive." Instead, it's to position yourself as the simpler, easier, case-specifically better solution for a certain type of customer.

Patio11 puts it well, as usual. Especially this paragraph:

"Is their product generic? Make yours hyper-specific. Is their product hyper-customized? Make yours the 'It only does 10% of what the other guy does, but it is the right 10%, and it actually works out of the box.'"

The good news is that $1B+ companies usually can't afford to target niches. Their divisions are expected to move the needle for, well, a $1B+ business. This puts enormous pressure on them to develop big, broad solutions that are maybe 75% adequate for as close to 100% of the market as they can reach. This is a strategic vulnerability that strongly favors the smaller player. It's one of the very few advantages a smaller player has in competition with a big player, particularly in the SaaS space.