Russia had agreed to ensure Syria's chemical weapons stockpiles were destroyed - and its failure to do this enabled the attack, he said.
G7 foreign ministers are preparing to meet in Italy later on Monday.
Talks will focus on how to increase pressure on Russia to distance itself from Syria's President Bashar Al-Assad.
On Tuesday, Mr Tillerson will continue from the G7 to Moscow, where he will meet his Russian counterpart, Sergei Lavrov.
Russia is the Syrian government's main ally, and helped facilitate a 2013 agreement to destroy Syria's chemical arsenal.
The suspected chemical attack on Khan Sheikhoun last Wednesday left 89 people dead.
In response, the US fired 59 missiles at a Syrian airbase on Friday.
Syria has denied using any chemical agents, and Russia says the US gave no evidence Syria has chemical weapons.
Analysis: BBC diplomatic correspondent James Robbins in Lucca, Italy
The next two days will be dominated by a collective search for arguments to persuade President Vladimir Putin he must now end Russia's military support for President Assad and help accelerate a negotiated political transition.
UK Foreign Secretary Boris Johnson is expected to press the case for new sanctions against Russia if they don't give ground. Mr Tillerson wants to go on from here to Moscow able to confront the Russians with a strong set of demands backed by key US allies.
But Mr Tillerson made clear over the weekend that Washington still regards the number one priority in Syria to be the elimination of so-called Islamic State.
Meanwhile, Russia and Iran, President Assad's key military backers, are threatening retaliation if there are any further American air strikes and the task of breaking the present deadlocks over Syria remains enormous.
Speaking on CBS's Face The Nation, Mr Tillerson said there was no evidence to suggest Russia was part of the alleged chemical attack.
However, "whether Russia was complicit here or whether they were simply incompetent or whether they got outwitted" by the Syrian government, they had "failed in their commitment to the international community", he said.
Russia had agreed to "be the guarantor of the destruction of Syria's chemical weapons stockpiles... the result of their failure has led to the killing of more children and innocents," Mr Tillerson added.
The Russian government has denied that Syria carried out a chemical attack, saying that civilians on the ground were poisoned after Syrian planes struck a depot producing chemical weapons for rebels.
Western governments, rebel leaders and a weapons expert all criticised this claim, saying the evidence pointed to an attack by the Syrian government.
Following the suspected attack, US President Donald Trump launched air strikes against a Syrian government air base, and branded Syrian leader Bashar al-Assad a "dictator" who had "launched a horrible chemical weapons attack on innocent civilians".
The Syrian government called the strikes "foolish and irresponsible", while Russia accused the US of "an act of aggression against a sovereign state delivered in violation of international law under a far-fetched pretext"