WASHINGTON — Along with its surprising military success, the Islamic State group has demonstrated a skill and sophistication with social media previously unseen in extremist groups.

And just as the United States has begun an aggressive air campaign against the militants, Richard A. Stengel, the under secretary of state for public diplomacy, believes the United States has no choice but to counter their propaganda with a forceful online response.

“Sending a jazz trio to Budapest is not really what we want to do in 2014,” said Mr. Stengel, referring to the soft-edged cultural diplomacy that sent musicians like Dave Brubeck on tours of Eastern-bloc capitals to counter communism during the Cold War. “We have to be tougher, we have to be harder, particularly in the information space, and we have to hit back.”

The State Department division that Mr. Stengel heads has tried a range of approaches for engaging with the Middle East since 9/11, from slick, Madison Avenue-style ads to traditional international-visitors and exchange programs.