Update React component by dispatching action from non-react component

I'm trying to update the cart-button on a project when the user adds an item to the cart.

I have build parts of the site in React.js - like the cart, the cart button etc.

configureStore.js:

export default function configureStore(initialState) {
      const store = createStore(
        reducers,
        initialState
      return store
    }

Action:

export function updateCart(payload) {
    return {
        type: CART_UPDATE,
        payload: payload
    }
}

Reducer:

export default function cart(state = {}, action) {

    switch (action.type) {

            case CART_UPDATE:

            const   cart = {
                        data: action.payload.cart,
                        errors: action.payload.errors,
                        isFetching: false
                    };

            return {
                ...state,
                ...cart
            };

    return state;
}

CartButton.js

... componnent etc.

function mapStateToProps(state) {
    return {
        cart: state.cart.data
    };
}

export default connect(mapStateToProps)(CartButton);

Provider

import configureStore from './store/configureStore'
var store = configureStore();

ReactDOM.render((<Provider store={store}><Cart showControls={true} /></Provider>), document.getElementById('react-cart'));

I'm dispatching an action that is supposed to update the cart quantity from a non-react component like this:

// imports 
import { dispatch } from 'redux';
import { updateCart } from '../../actions/cart_actions';
import configureStore from '../../store/configureStore'
var store = configureStore();

and then..

store.dispatch(updateCart(response));

The action is dispatched and the state is updated. The cart-button component is connected via. react-redux connect() function. But somehow it isn't updating the component with the new quantity.

When I dispatch an action from within my cart React component it works fine.

I might be missing something obvious. Any suggestions?