They said Haley is one of the most disciplined and cautious members of the Cabinet, especially when it comes to her public appearances.
Three sites were hit - the Barzeh complex, which the United States says is a centre for development, production and testing of chemical and biological weapons, although Syria denies this.
The Trump administration is weighing a new round of economic sanctions against Russian Federation for its backing of Syrian President Bashar al Assad's alleged use of chemical weapons.
Russia's deputy foreign minister said the delay was due to US airstrikes.
Haley also said that although "it is all of our goal to see American troops come home", the United States won't leave Syria before accomplishing President Donald Trump's three major goals: eliminating the threat of chemical weapons attacks in Syria, defeating ISIS "completely and wholly", and making "sure that we had good grounds to watch what Iran was doing".
The next task was convincing Trump that he should punish Putin in coordination with the Europeans.
For the near future, the strikes suggest that the White House has no intention of withdrawing 2,000 US troops now in Syria, as Trump suggested on April 4 that he intends to do.
"Right now they don't have very good friends and right now the friends that they do have are causing them harm", Haley said, referring to Russian Federation.
"I'm afraid the Syrian war in many ways will go on in its awful, miserable way". "But be very clear, if we leave, when we leave, it will be because we know that everything is moving forward".
The watchdog, the OPCW (Organisation for the Prohibition of Chemical Weapons), has been meeting behind closed doors in The Hague to discuss the alleged attack on 7 April.
The U.S. ambassador to the OPCW, Kenneth Ward, said there were suspicions that Russian Federation, a longtime ally of the Syrian government, may have tampered with the evidence. "It is our concern that they may have tampered with it with the intent of thwarting the efforts of the OPCW fact-finding mission to conduct an effective investigation".
The Swedish delegation cited a briefing from OPCW director-general that said Syria and Russian Federation were concerned that security on the ground could not be guaranteed. It also said Russian officers found no patients with chemical attack symptoms at a local hospital, and no indication of any burials having taken place of the victims.
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A second USA official said targets were being carefully selected with the aim of damaging Assad's ability to conduct further gas attacks, while avoiding the risk of spreading poisonous fallout in civilian areas.
Putin said the actions violated the U.N. Charter and if they continue, "it will inevitably entail chaos in worldwide relations", the statement said.
According to Hamilton 68, a website set up by researchers working with the Alliance for Securing Democracy, a transatlantic project set up to counter Russian disinformation campaigns - Syria is the leading hashtag on social media accounts identified as part of Russia-linked influence networks.
Moscow blamed the delay on the Western air strikes.