“At a minimum, this kind of action carries the risk of dragging Yemen into a regional conflagration,” Mr Griffiths told the UN Security Council.
“Of one thing we can be certain, that this very serious incident makes the chances of a regional conflict that much higher, with Yemen in some way linked. None of that is good for Yemen.”
Houthi rebels, who have been fighting government forces in Yemen since late 2014 when they seized the capital Sanaa, claimed responsibility for the attacks in Saudi Arabia.
The Iran-backed rebels, who are backed by Iran, have been conducting such attacks for years.
But the US blamed Iran for the latest drone attacks, saying there was no evidence that strikes on Abqaiq, the world's largest oil processing plant, and the Khurais oilfield had come from Yemen.
The damage resulted in a major cut to global supplies of crude oil.
Mr Griffiths, speaking by video link from Amman, said that regardless of who was behind the strikes “the fact that Ansar Allah [the Houthis] claimed responsibility is bad enough”.
“Whatever we will discover of the attack it is a sure sign that Yemen seems to be moving even further away from the peace we all seek,” he said.
Mr Griffiths has for almost two years led the international push for peace in Yemen, holding talks with government officials and the Houthis.
US ambassador to the UN Kelly Craft repeated Secretary of State Mike Pompeo's claim that Iran was responsible, but other members of the Security Council said they would investigate before assigning blame.
The UK's permanent representative to the UN, Karen Pierce, called the attacks “reckless, despicable and totally unjustifiable”, and said there was a great danger of letting the Yemen situation slide backwards.
Instability in Yemen has deepened in recent months with the secessionist South Transitional Council taking control of government buildings in Aden.
Mr Griffiths urged all sides to embrace a Saudi-led reconciliation effort.
“I welcome the presence of delegations from the government of Yemen and Southern Transitional Council in Jeddah,” he said.
“I know this council joins me to wish the Saudis and the two delegations every success.”
The humanitarian situation in Yemen also remains dire.
Mark Lowcock, the UN's undersecretary general for humanitarian affairs, told the council on Monday that the environment for aid agencies “has perhaps never been worse”.
Most incidents “are due to restrictions imposed by Houthi authorities”, Mr Lowcock said.
He said there were 300 incidents in June and July that hindered humanitarian assistance.