MOSCOW — Ukraine’s new president, Volodymyr Zelensky, was on track Monday to become the country’s first post-Soviet chief to enjoy an outright parliamentary majority — potentially a watershed moment for an untested leader who has vowed to take on endemic corruption.

Mr. Zelensky’s party, Servant of the People — named for the television comedy in which he played a teacher who unexpectedly becomes president — appeared to have won more than 250 of the 424 seats in Parliament in Sunday’s election.

Speaking after exit poll data was released Sunday night, Mr. Zelensky, 41, said the results put a “huge responsibility” on him to deliver on his promises, including an end to the five-year-old war with Russian-backed separatists in eastern Ukraine that has claimed more than 13,000 lives.

“We won’t fail the Ukrainian people,” Mr. Zelensky said, speaking to supporters at his party’s campaign headquarters. “Our main priorities, just as for any Ukrainian, I am sure, are ending the war, return of our hostages and victory over corruption.”