By Tim Higgins

Apple Inc. countersued Epic Games Inc. on Tuesday, accusing the "Fortnite" maker of breaching a contract when it introduced a new in-app payment system within the game, as the battle between the iPhone company and software developer continues to intensify.

Apple is asking a judge to award punitive damages and restrict Epic from continuing what it describes as unfair business practices. The filing came after Epic sought an injunction on Friday to force Apple to put the popular videogame back in its App Store.

"Epic's lawsuit is nothing more than a basic disagreement over money," Apple said in its filing Tuesday. "Although Epic portrays itself as a modern corporate Robin Hood, in reality it is a multi-billion dollar enterprise that simply wants to pay nothing for the tremendous value it derives from the App Store."

An Epic spokesman declined to comment beyond what the company has said previously about its efforts to fight what it sees as a monopoly payment system that suppresses competition and inflates prices.

The dispute began when Epic introduced the payment system on Aug. 13 within the "Fortnite" app that skirted the 30% fee that Apple and Alphabet Inc.'s Google charge for in-app purchases. Within hours, both companies removed the game from their app marketplaces for violating their rules. Epic later sued Apple and Google in California federal court, accusing both companies of anticompetitive conduct.

Epic is just the latest company to take issue with Apple's control of the App Store, which is part of the Cupertino, Calif.-company's evolution to expand beyond selling hardware and to generate revenue from software services. Apple's App Store in the first seven months of this year generated nearly $39 billion in global revenue from in-app purchases, subscriptions and premium apps, according to researcher Sensor Tower.

Its App Store has about 1 billion customers from 175 countries. "Fortnite" has been downloaded almost 130 million times through the App Store since being added to it in 2018, according to Apple.

In its filing Tuesday, Apple sought to portray itself as a victim of plotting by Epic. The tech giant described all the help it had given Epic in recent years and detailed how it alleged the game company had begun laying groundwork for what Apple called a "calculated and pre-packaged campaign" against it.

Epic Chief Executive Tim Sweeney in late June emailed Apple executives, apparently including Chief Executive Tim Cook, asking for a special agreement exempting his company from existing contractual obligations, including the App Store payments, according to the filing. Apple rejected that demand.

Then on Aug. 3, Epic sent to the App Store an update of "Fortnite" that the game company described as a "hotfix" but that Apple claimed was a Trojan horse allowing it to bypass Apple's review process and payment system.

At around 2 a.m. on Aug. 3, Mr. Sweeney emailed Apple again. "Epic will no longer adhere to Apple's payment processing restrictions," he wrote, according to the filing.

A few hours later, Apple said, Epic activated the external payment workaround that cut Apple out of its share of sales. "This was little more than theft," Apple wrote. "Epic sought to enjoy all of the benefits of Apple's iOS platform and related services while its 'hotfix' lined Epic's pockets at Apple's expense."

That was followed by a public relations battle and lawsuit against Apple. Epic showed a video on "Fortnite" mimicking the iPhone-maker's '1984' ad campaign.

Apple has defended its take of app purchases in the past as a way to help fund development of its ecosystem that provides a safe and secure way to download third-party software. It said in the new filing that Apple requires every app on its system to undergo a "rigorous, human-assisted review."

Epic has earned more than $600 million from its relationship with Apple, according to court records.

Apple noted that Epic has its own app marketplace where it charges a commission as well. "Epic's intention is thus straightforward: It seeks free access to the Apple-provided tools that it uses and -- worse yet -- it wishes to then charge others for access to Apple's intellectual property and technologies," the tech giant said.

On Friday, Epic reiterated its arguments accusing Apple of monopolistic behavior, an accusation the tech giant pushed back on Tuesday, noting that "Fortnite" is available on six different mobile, computer and game-console platforms.

"There is nothing anticompetitive about charging a commission for others to use one's service," Apple said in the filing.

Apple is facing antitrust probes from governments in the U.S. and Europe while other tech companies have joined in complaining publicly about its power. Last month, Facebook Inc. CEO Mark Zuckerberg said in a webcast with employees that Apple has the ability to block innovation and "to charge monopoly rents," according to a person familiar with the matter.

Write to Tim Higgins at Tim.Higgins@WSJ.com