The gravitational pull the Sun has on the Moon is roughly two times greater than the pull the Earth has on the Moon. If this is the case, why does the Sun's gravity not pull the Moon away from the Earth?
The Sun pulls on the Earth as well. So both Earth and Moon are "falling towards the Sun" all the time, just as they are moving in almost the same orbit.
Earth causes the orbit of the Moon to "wobble" a little bit. If you were simply given the coordinates of the Moon as it moves around the Sun, you would notice there is a deviation from the expected ellipse - which tells you there is "something else" there. But the Moon never goes into retrograde motion - in other words, it "keeps going forward" in its orbit, with just a little wobble.
The fact that we think the Earth is "holding on" to the Moon comes from our geocentric view of the universe.