Start a build and passing variables through Azure DevOps Rest API

Variables are included in definitions, you can update your build definition to set the variables via build-definition api first and then queue the build.

Following is the variable section get via build-definition api:

  "variables": {
    "system.debug": {
      "value": "false",
      "allowOverride": true
    },
    "BuildConfiguration": {
      "value": "release",
      "allowOverride": true
    },
    "BuildPlatform": {
      "value": "any cpu",
      "allowOverride": true
    }
  },

For anyone having problems with this (I did), there is a difference in APIs used since the accepted answer (which to me didn't work at all). But following Cyprien Autexier's advice, I took a look under the hood (Firefox Dev Tools) and I noticed the portal does not use the Builds API anymore. It uses the Pipelines one (https://docs.microsoft.com/en-us/rest/api/azure/devops/pipelines/runs/run-pipeline?view=azure-devops-rest-6.1). With this, worked flawlessly.

The accepted answer does not really answers the question when you need to set a value at queue time. The solution is actually pretty simple you just have to add a parameters field to the json payload. The content should be a json string (not directly an object) containing the parameters Ex :

{
    "parameters":  "{\"ReleaseNumber\":  \"1.0.50\", \"AnotherParameter\":  \"a value\"}",
    "definition":  {
                       "id":  2
                   }
}

EDIT : This feature is now properly documented as an optional stringified dictionary. See https://www.visualstudio.com/fr-fr/docs/integrate/api/build/builds#queue-a-build