Estou tentando o novoSuporte MultiDex no meu aplicativo e até agora consegui compilar meu aplicativo corretamente, mas ao executá-lo, recebo a seguinte exceção:

java.lang.RuntimeException: Unable to instantiate application android.support.multidex.MultiDexApplication: java.lang.ClassNotFoundException: Didn't find class "android.support.multidex.MultiDexApplication" on path: DexPathList[[zip file "/data/app/me.myapp.main-2.apk"],nativeLibraryDirectories=[/data/app-lib/me..main-2, /vendor/lib, /system/lib]]
    at android.app.LoadedApk.makeApplication(LoadedApk.java:507)
    at android.app.ActivityThread.handleBindApplication(ActivityThread.java:4382)
    at android.app.ActivityThread.access$1500(ActivityThread.java:139)
    at android.app.ActivityThread$H.handleMessage(ActivityThread.java:1270)
    at android.os.Handler.dispatchMessage(Handler.java:102)
    at android.os.Looper.loop(Looper.java:136)
    at android.app.ActivityThread.main(ActivityThread.java:5086)
    at java.lang.reflect.Method.invokeNative(Native Method)
    at java.lang.reflect.Method.invoke(Method.java:515)
    at com.android.internal.os.ZygoteInit$MethodAndArgsCaller.run(ZygoteInit.java:785)
    at com.android.internal.os.ZygoteInit.main(ZygoteInit.java:601)
    at dalvik.system.NativeStart.main(Native Method)
Caused by: java.lang.ClassNotFoundException: Didn't find class "android.support.multidex.MultiDexApplication" on path: DexPathList[[zip file "/data/app/me.myapp.main-2.apk"],nativeLibraryDirectories=[/data/app-lib/me.myapp.main-2, /vendor/lib, /system/lib]]
    at dalvik.system.BaseDexClassLoader.findClass(BaseDexClassLoader.java:56)
    at java.lang.ClassLoader.loadClass(ClassLoader.java:497)
    at java.lang.ClassLoader.loadClass(ClassLoader.java:457)
    at android.app.Instrumentation.newApplication(Instrumentation.java:998)
    at android.app.LoadedApk.makeApplication(LoadedApk.java:502)

Este é o meu arquivo gradle:

apply plugin: 'com.android.application'

android {
    compileSdkVersion 21
    buildToolsVersion "21.1.0"

    defaultConfig {
        minSdkVersion 16
        targetSdkVersion 21
        multiDexEnabled true
    }
    buildTypes {
        release {
            minifyEnabled true
            proguardFiles getDefaultProguardFile('proguard-android.txt'), 'proguard-rules.txt'
        }
        debug {
            minifyEnabled false
            proguardFiles getDefaultProguardFile('proguard-android.txt'), 'proguard-rules.txt'
        }
    }
    packagingOptions {
        exclude 'META-INF/LICENSE.txt'
        exclude 'META-INF/NOTICE.txt'
    }
}

repositories {
    mavenCentral()
    mavenLocal()
}


dependencies {
    compile 'com.android.support:multidex:1.0.0'
    compile fileTree(include: ['*.jar'], dir: 'libs')
    compile 'com.android.support:support-v4:21.0.0'
    compile 'com.android.support:support-v13:21.0.0'
    compile 'com.android.support:appcompat-v7:21.0.0'
}

E meu AndroidManifest.xml:

<application
android:name="android.support.multidex.MultiDexApplication">

Eu não entendo qual é o problema. Eu acho que estou fazendo tudo de acordo com a documentação. Há algo mais que estou perdendo? Verifiquei se tinha a biblioteca de suporte mais recente e o repositório instalado a partir do gerenciador do SDK.