This article shows you how to move your Custom List from SharePoint On-Premise to Office 365 SharePoint Online.
In the actual fact you can use the same way to export and import the Custom List between Office 365 SharePoint Online,
This step-by-step article has 2 parts.
1. Download the List template.
2. Upload the List template.
Download the List template
1. Open the Custom List you want to Export.
2. Click List on the Ribbon.
3. Click List Settings.
4. Choose Save list as template under the Permissions and Management.
5. Give the list a file name then click OK.
6. Click the Gear on the right top and choose Site Settings.
7. Under the Web Design Galleries, click List templates.
8. Click FILES on the ribbon.
9. Click Download a Copy.
10. Give a file name to the stp file and click OK.
Upload the List template
Login to Office 365 SharePoint that you want to upload the Custom List.
1. Go to Site Settings (same as step 6 above).
2. Under the Web Design Galleries, click List templates (same as step 7 above).
3. Click on Upload Document.
4. Select the stp file then click Save.
Important: The default Site Collection you have you might not see Web Design Galleries (in my case). Create a new Site Collection.