To understand what is going on in Romans 8:1-11, it will be helpful to have two passages from Jeremiah and Ezekiel in mind. They say this:

“The days are coming,” declares the Lord,
“when I will make a new covenant
with the people of Israel…
This is the covenant I will make with the people of Israel
after that time,” declares the Lord.
“I will put my law in their minds
and write it on their hearts.
I will be their God,
and they will be my people.
No longer will they teach their neighbor,
or say to one another, ‘Know the Lord,’
because they will all know me,
from the least of them to the greatest,”
declares the Lord.
“For I will forgive their wickedness
and will remember their sins no more.”
– Jeremiah 31:31-34

For I will take you out of the nations; I will gather you from all the countries and bring you back into your own land. I will sprinkle clean water on you, and you will be clean; I will cleanse you from all your impurities and from all your idols. I will give you a new heart and put a new spirit in you; I will remove from you your heart of stone and give you a heart of flesh. And I will put my Spirit in you and move you to follow my decrees and be careful to keep my laws.
– Ezekiel 36:24-27

Both of these hope-filled passages were given to the people of Israel when they were suffering the punishment for their sin. Though they make their point in different ways, both Jeremiah and Ezekiel say that God will (1) forgive/cleanse the people’s sin and (2) do an inner work by his Spirit in the people that will empower them to do God’s will. Regarding that second piece, we might say that God will overcome the power of sin in the people’s lives. As Jeremiah says, this work will happen when God makes a “new covenant” with his people.

Here’s the thing you need to know about the new covenant – Jesus attached it to himself. In Luke’s Gospel, Jesus says this concerning the cup at the last supper:

This cup is the new covenant in my blood, which is poured out for you.
– Luke 22:20, emphasis mine

The work of Christ has brought about the new covenant, which means that God has done the dual work of forgiving and overcoming sin in Christ!

When we come to Romans 8:1-11, this dual work of the new covenant is on display. God has presented Christ as a “sin offering” and set people free from the law of sin and death through the Spirit. So much of where we have come from in Romans has been describing a previous era when we were powerlessness before sin. Indeed Romans 8:6-8 tells us as much when it speaks of the flesh:

The mind governed by the flesh is death … Those who are in the realm of the flesh cannot please God.

Thankfully, verse 9 comes quick on its heels.

You, however, are not in the realm of the flesh but are in the realm of the Spirit, if indeed the Spirit of God lives in you.

The new covenant has arrived, complete with its inner work of the Spirit. What is left for believers is to align ourselves with the work of the Spirit in our lives. As we do, we put sin to death and live more and more to and for God.