Finn, who is now in isolation, revealed he is experiencing "mild symptoms".
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"I am relieved to be only experiencing mild symptoms at this stage and grateful for the protection that my triple boosted vaccination is providing.
"I am so sorry to disappoint our fans this coming week, but rest assured we will be back to make good."
The rest of the band's Dreamers Are Waiting tour, which was due to end with two A Day On The Green festivals this weekend, will be rescheduled soon.
Crowded House were set to play at Hobart's Mystate Bank Arena tonight, Canberra's Royal Theatre on April 21, and A Day On The Green dates at Mt Duneed Estate, Geelong on April 23, and Bimbadgen, Hunter Valley on April 24.