Yes the Godhead works together to draw people to God and sanctify and save them.
Christ suffered and died and bought us salvation. The Father draws us to Christ (presumably through the urging of the Holy Spirit.) Then, Christ lives in us and sanctifies us by the Holy Spirit to help us do the Will of the Father. So we see that all of God works together for our salvation.
Christ is the only way to the Father who is God because Christ is God. To know Him is to know the Father.
John 14:7 says: If you really know me, you will know my Father as well. From now on, you do know him and have seen him.
Take a look at Romans 8:29-30:
29 “For those God foreknew he also predestined to be conformed to the image of his Son, that he might be the firstborn among many brothers and sisters.”
30 “And those he predestined, he also called; those he called, he also justified; those he justified, he also glorified.”
God knew us before we were born (Jeremiah 1:5) and those that He knows will accept Christ, He has predestined to be drawn to Christ. He calls us to Christ by the Holy Spirit, and justifies us by Christ’s sacrifice as well as sanctifying us by our obedience to the guiding of the Holy Spirit. And He will glorify us at our resurrection as we will take part in Christ’s glory, because He has justified us.