Tillerson is due to meet Lavrov in Moscow next week.
The United States' top diplomat also said it was "important that we keep our priorities straight" and the USA believes "the first priority is the defeat of ISIS".
"I'm disappointed in that response from the Russians because it indicates their continued support for the Assad regime", said Tillerson, who is expected to make a hard trip to Moscow next week.
US Secretary of State Rex Tillerson expressed disappointment with Russia's negative reaction to the missile strikes in Syria, but added that Moscow's support to the regime of President Bashar Assad is not surprising.
Tillerson stopped short of accusing the Russians of complicity in a suspected sarin nerve gas attack April 4 that killed at least 87 civilians in Syria's southern Idlib province.
In The Sunday Times, Sir Michael said Russian Federation needed to take action to make sure the "needless suffering" of Syrian civilians can be brought to an end.
The decision comes after a dramatic downturn in relations between American and Russian Federation after a chemical attack killed more than 70 Syrians, including children.
"We don't believe we need dialogue with London more than (London) needs it (with us)".
Tillerson's comments on the role of Assad have shifted in recent days.
Other than that, he added, "there is no change to our military posture" towards Syria.
"What happened, of course, largely sets back our cooperation" on Syria, Russian Deputy Foreign Minister Gennady Gatilov was quoted as saying by Interfax.
"If Russia wants to be absolved of responsibility for future attacks, Vladimir Putin needs to enforce commitments, to dismantle Assad's chemical weapons arsenal for good, and to get fully engaged with the United Nations peacekeeping progress".
US Ambassador to the UN Nikki Haley meanwhile suggested that while the fight against the Islamic State group was a top priority, securing Assad's removal from power was also an American priority.
But sources insisted he has always believed Assad can not remain in power. "If you look at his actions, if you look at the situation, it's going to be hard to see a government that's peaceful and stable with Assad".