Christ's Lordship is bound up in His being the Savior. Here is why:
When a person comes to faith in Christ as Savior, are they not acknowledging their helplessness, and so come to Him in the hope that He will free them from both the guilt and power of sin? Do not people come to Christ so that He might liberate them from sin's tyranny over them? In coming to Christ as Savior from sin we implicitly acknowledge His Lordship, for we no longer want to be under the rule of sin. When the Holy Spirit opens our eyes to our wretched condition before God, we want to be free of sin but, knowing we are impotent to save ourselves from it, we ask Christ to free us. No one who is truly converted comes to Christ and says "please don't free me from sin". Anyone who did this is not even coming to him as Savior but hoping He will leave them in their original condition as slaves.
We are all painfully aware of the remainders of sin that exist in us as believers. But if the Holy Spirit dwells in us, we are incapable of remaining indifferent to it. We will mourn over it and desire to be rid of it, but being unable to do so ourselves, we come to the Savior daily plea for the help we so desperately need. Can a person claim to believe in Christ as their Savior and be content to remain in sin? Will the Holy Spirit allow that? (1 Cor 11:31-32) If so then in what sense is Christ their Savior?