The word "drew" is of course the past tense of "draw", as "pulled" is of "pull".
The word "Draw" has quite a few meanings as a verb (and yet others as other parts of speech) as seen in this definition from Merriam-Webster (and there are other meanings I can think of not listed in this entry)
In this sense the literal meaning of sense 1 from the linked definition
to cause to move continuously toward or after a force applied in advance : pull
or the figurative meaning of sense 3
to bring by inducement or allure : attract
"Pull" has sufficiently similar meanings that it could easily be substituted here, the choice is one of style.