To cleanse the house then, he shall take two birds and cedar wood and a scarlet string and hyssop, and he shall slaughter the one bird in an earthenware vessel over running water. Then he shall take the cedar wood and the hyssop and the scarlet string, with the live bird, and dip them in the blood of the slain bird as well as in the running water, and sprinkle the house seven times. He shall thus cleanse the house with the blood of the bird and with the running water, along with the live bird and with the cedar wood and with the hyssop and with the scarlet string. However, he shall let the live bird go free outside the city into the open field. So he shall make atonement for the house, and it will be clean. (Leviticus 14:49-53)
Leviticus 16 speaks of the high priest’s duty on Yom Kippur (the Day of Atonement) and the cleansing due to impurities:
Then he shall slaughter the goat of the sin offering which is for the people, and bring its blood inside the veil and do with its blood as he did with the blood of the bull, and sprinkle it on the mercy seat and in front of the mercy seat. He shall make atonement for the holy place, because of the impurities of the sons of Israel and because of their transgressions in regard to all their sins; and thus he shall do for the tent of meeting which abides with them in the midst of their impurities. When he goes in to make atonement in the holy place, no one shall be in the tent of meeting until he comes out, that he may make atonement for himself and for his household and for all the assembly of Israel. Then he shall go out to the altar that is before the Lord and make atonement for it, and shall take some of the blood of the bull and of the blood of the goat and put it on the horns of the altar on all sides. With his finger he shall sprinkle some of the blood on it seven times and cleanse it, and from the impurities of the sons of Israel consecrate it. (Leviticus 16:15-19)
Note that the atonement is primarily personal ("that he may make atonement for himself") then familial ("and for his household") and finally national ("and for all the assembly of Israel").
Ezekiel 43 speaks of the future time when the presence of G-d returns to the Temple (Ezekiel 43:1-2) and again commands an offering for cleansing:
And He said to me, "Son of man, thus says the Lord God, 'These are the statutes for the altar on the day it is built, to offer burnt offerings on it and to sprinkle blood on it. You shall give to the Levitical priests who are from the offspring of Zadok, who draw near to Me to minister to Me,' declares the Lord God, 'a young bull for a sin offering. You shall take some of its blood and put it on its four horns and on the four corners of the ledge and on the border round about; thus you shall cleanse it and make atonement for it. You shall also take the bull for the sin offering, and it shall be burned in the appointed place of the house, outside the sanctuary.
'On the second day you shall offer a male goat without blemish for a sin offering, and they shall cleanse the altar as they cleansed it with the bull. When you have finished cleansing it, you shall present a young bull without blemish and a ram without blemish from the flock. You shall present them before the Lord, and the priests shall throw salt on them, and they shall offer them up as a burnt offering to the Lord. For seven days you shall prepare daily a goat for a sin offering; also a young bull and a ram from the flock, without blemish, shall be prepared. For seven days they shall make atonement for the altar and purify it; so shall they consecrate it. When they have completed the days, it shall be that on the eighth day and onward, the priests shall offer your burnt offerings on the altar, and your peace offerings; and I will accept you,' declares the Lord God." (Ezekiel 43:18-27)
The apostolic affirmation of this "atonement = cleansing" concept is found in Hebrews chapter 9:
But when Christ appeared as a high priest of the good things to come, He entered through the greater and more perfect tabernacle, not made with hands, that is to say, not of this creation; and not through the blood of goats and calves, but through His own blood, He entered the holy place once for all, having obtained eternal redemption. For if the blood of goats and bulls and the ashes of a heifer sprinkling those who have been defiled sanctify for the cleansing of the flesh, how much more will the blood of Christ, who through the eternal Spirit offered Himself without blemish to God, cleanse your conscience from dead works to serve the living God? (Hebrews 9:11-14)
The author of Hebrews affirms that the blood of goats and bulls (and the ashes of the red heifer) cleanses the flesh and then uses that truth in an a fortiori (Hebrew: kal va-chomer) argument to prove "how much more" the blood of Christ cleanses our conscience.
Since the former is true then the latter is much more!
One final thought: although the Hebrew words for atoning or atonement are used over 100 times in the Old Testament, the words "atone", "atoning", or "atonement" are never found in the NASB translation of the New Testament. The Greek word used in the LXX for "atoning" (hilaskomai) is only used twice: by the publican in Luke 18:13 and in Hebrews 2:17 where it is, inaccurately in my opinion, translated as "propitiation" (an appeasement). Scripture never says that Messiah is our atonement. It says He made atonement... with His own blood.
Blessed be His precious Name.