I would argue that this assertion is only partly true. Yes, Donne feels that his soul has been captured by the Satanic desires of the selfish heart. But that doesn't mean that he's sold his soul to the Devil, nor does it mean that reason is inherently Satanic. After all, Donne explicitly refers to reason as "your viceroy in me," clearly implying that reason is a gift given to human beings by the Almighty.
It's not so much, then, that reason is Satanic; it's that it is inadequate on its own to liberate the speaker's soul from its selfish desires. In keeping with the poem's central conceit of the speaker's heart as a besieged town, God's viceroy—or representative—has been overwhelmed by dark forces. It's as if reason has been sent by God as an ambassador into the besieged fortress of the speaker's heart, only to find himself incapable of driving out the Satanic desires that have taken root there.
That being the case, there's only one thing for it. God needs to come and ram down the doors of the speaker's heart to liberate him from sin. On this reading, powerful negative emotions—emotions steeped in sin—cannot be expelled by reason but by powerful positive emotions: the kind induced by God's love.