Russia launched air strikes in late September in support of Mr Assad, Moscow's ally.
Mr Assad told Czech television that US air strikes on Syria had not slowed IS, but that Russian bombing was doing so.
He also condemned the recent shooting down of a Russian warplane by Turkey.
Russia has been bombing rebel positions in Syria, but said on launching the air strikes that its main target was IS.
The US said last month that most of the air strikes were against "moderate Syrian opposition forces" opposed to Syria's government.
Mr Assad said that since US air strikes on Syria began in September last year, IS "has expanded and the recruiting from around the world has increased".
He added: "Since the participation of Russia in the same fight (...) Isis has been shrinking - and al-Nusra Front of course and the other terrorist groups. The facts are telling."
However, in September, before Russia began its sorties, the US defence department said IS had already lost up to 10% of its territory in Syria in a year.
On Wednesday, US Secretary of State John Kerry said Russia's role in Syria was welcome, as long as Moscow was focused only on IS.
"They can be an extremely constructive and important player in reaching a solution," he said after talks between Nato members in Brussels.
Nato members were ready to step up their role against IS, Mr Kerry said.
Also in the Czech television interview, Mr Assad also criticised the government of Syria's neighbour, Turkey, after its shot down a Russian warplane inside Syria on 24 November.
In the interview, Mr Assad said the incident showed that his Turkish counterpart, Recep Tayyip Erdogan, had "lost his nerve" because Russia's involvement had altered what was at stake.
"The failure of Erdogan in Syria, the failure of his terrorist groups, means his political demise," he said.
Syria's president Bashar al-Assad says Russia's involvement in his country's war has led to significant changes, including the "shrinking" of the Islamic State (IS) group.