Make sure you have the following extension installed on VS Code as I think it would prove help. Make sure your project is structured as follows. When you use the extension to observe your development environment, you'll notice the host.json, connections.json and workflow.json under folder with a corresponding workflow name.
You've stated that you've already configured deployment center with your Azure DevOps repo, so assuming your project has the above structure, need to create a build task that will deploy the zip file to the logic app.
- task: ArchiveFiles@2
displayName: "Archive files"
inputs:
rootFolderOrFile: "$(System.DefaultWorkingDirectory)/publish_output"
includeRootFolder: false
archiveFile: "$(System.DefaultWorkingDirectory)/build$(Build.BuildId).zip"
- task: PublishBuildArtifacts@1
inputs:
PathtoPublish: '$(System.DefaultWorkingDirectory)/build$(Build.BuildId).zip'
artifactName: 'drop'
- task: AzureFunctionApp@1
displayName: 'Deploy logic app workflows'
inputs:
azureSubscription: 'MyServiceConnection'
appType: 'functionAppLinux' ## Default: functionApp
appName: 'MyLogicAppName'
package: $(System.ArtifactsDirectory)/**/*.zip
deploymentMethod: 'zipDeploy'