How does the Moon's gravity affect Earth's oceans despite Earth's stronger gravitational pull? Given that Earth has a much stronger gravitational pull than the Moon, how does the Moon have any influence on Earth's oceans?
Everything in the universe has a gravitational influence on everything else in the universe. It isn't a question of the strongest gravitational pull winning out and all the others doing nothing.

The Earth is the strongest pull on the oceans, but the Moon and the Sun both have easily measurable effect in addition to the Earth's. Other bodies (Venus, Jupiter,  a small asteroid in another galaxy,....) all have much smaller effects which will be hard or impossible to detect amidst the noise due to waves and so on.