$quote = $this->quoteRepository->getActive($cartId);

    // Extension Attributes
    $quoteExtension = $quote->getExtensionAttributes();
    $quoteExtension->setFromApp(true);

    $this->quoteRepository->save($quote);

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Api/etc/extension_attributes.xsd">
<extension_attributes for="MagentoQuoteApiDataCartInterface">
    <attribute code="from_app" type="string"/>
</extension_attributes>

<?xml version="1.0" encoding="UTF-8"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:ObjectManager/etc/config.xsd">
<type name="MagentoQuoteApiCartRepositoryInterface">
    <plugin name="vendor_namespace_plugin_cart_repository" type="VendorNamespacePluginCartRepositoryPlugin"/>
</type>

$quote->setFromApp(true);
$this->quoteRepository->save($quote);

$quote = $this->quoteRepository->getActive($cartId);

// Extension Attributes
$quoteExtension = $quote->getExtensionAttributes();
$quoteExtension->setFromApp(true);
$quote->setExtensionAttributes(quoteExtension);

$this->quoteRepository->save($quote);

<?php
namespace VendorNamespacePlugin;

use MagentoQuoteApiCartRepositoryInterface;
use MagentoQuoteApiDataCartExtension;
use MagentoQuoteApiDataCartExtensionFactory;
use MagentoQuoteApiDataCartInterface;
use MagentoQuoteApiDataCartSearchResultsInterface;

/**
 * Class CartRepositoryPlugin
 */
class CartRepositoryPlugin
{
    /**
     * @var CartExtensionFactory
     */
    private $extensionFactory;

    /**
     * CartRepositoryPlugin constructor.
     *
     * @param CartExtensionFactory $orderExtensionFactory
     */
    public function __construct(
        CartExtensionFactory $orderExtensionFactory
    ) {
        $this->extensionFactory = $orderExtensionFactory;
    }

    /**
     * @param CartRepositoryInterface $subject
     * @param CartInterface $resultEntity
     * @return CartInterface
     * @SuppressWarnings(PHPMD.UnusedFormalParameter)
     */
    public function afterGet(
        CartRepositoryInterface $subject,
        CartInterface $resultEntity
    ) {
        /** @var CartExtension $extensionAttributes */
        $extensionAttributes = $resultEntity->getExtensionAttributes() ?: $this->extensionFactory->create();

        $extensionAttributes->setFromApp($resultEntity->getData('from_app'));

        $resultEntity->setExtensionAttributes($extensionAttributes);

        return $resultEntity;
    }

    /**
     * @param CartRepositoryInterface $subject
     * @param CartSearchResultsInterface $resultCart
     * @return CartSearchResultsInterface
     */
    public function afterGetList(
        CartRepositoryInterface $subject,
        CartSearchResultsInterface $resultCart
    ) {
        /** @var CartInterface $order */
        foreach ($resultCart->getItems() as $order) {
            $this->afterGet($subject, $order);
        }

        return $resultCart;
    }

    /**
     * @param CartRepositoryInterface $subject
     * @param CartInterface $result
     * @return array
     */
    public function beforeSave(
        CartRepositoryInterface $subject,
        CartInterface $quote
    ) {
        $extensionAttributes = $quote->getExtensionAttributes() ?: $this->extensionFactory->create();
        if ($extensionAttributes !== null && $extensionAttributes->getFromApp() !== null) {
            $quote->setFromApp($extensionAttributes->getFromApp());
        }

        return [$quote];
    }
}

/**
 * @param CartRepositoryInterface $subject
 * @param CartInterface $result
 * @return CartInterface
 */
public function afterSave(
    CartRepositoryInterface $subject,
    CartInterface $result
) {
    $extensionAttributes = $result->getExtensionAttributes() ?: $this->storeExtensionFactory->create();
    if ($extensionAttributes !== null && $extensionAttributes->getFromApp() !== null) {
        /** @var CustomEntity $customEntity */
        $customEntity = $this->customEntityFactory->create();
        $customEntity->setQuoteId($result->getId());
        $customEntity->setFromApp($result->getFromApp());
        $this->customEntityRepository->save($customEntity);
    }

    return $result;
}