There was speculation that leading Tories would put pressure on their MPs to vote against the probe being carried out by the Privileges Committee in a bid to delay the process and protect Mr Johnson.
The issue is important because government regulations dictate that a minister who knowingly misleads the Commons would be expected to resign.
Labour leader Sir Keir Starmer this week called on the PM to quit.
Mrs Kearns posted on her Facebook page yesterday (Thursday): “Earlier this week I informed the government that I would not support them if they whipped us to vote against an investigation by Parliament's Privilege Committee into whether the Prime Minister misled Parliament.
"I'm pleased that the Prime Minister has listened to me and my colleagues and decided not to oppose the investigation."