Ivanna Klympush-Tsintsadze told BBC Scotland other countries could impose more serious sanctions and there were "plenty of opportunities to help Ukraine”.
Her comments come as Ukrainian president Volodymyr Zelensky said 137 Ukrainian citizens – both soldiers and civilians – died on Thursday following a full-scale Russian invasion launched by Mr Putin.
The United Nations has reported 100,000 people in the capital Kyiv and elsewhere have fled.
Ms Klympush-Tsintsadze told BBC Good Morning Scotland there had been fighting in the streets and blasts heard across Kyiv with civilian buildings hit.
She said: “I am counting on our military.
"We Ukrainians are fighters and are doing our best on our own territory in order to protect our city and our country.”
The former deputy prime minister said: "I think the world should do much more serious sanctions imposed on the Russian Federation.” She said “immediate reactions and sanctions” were “crucial”.
Ms Klympush-Tsintsadze added: "Putin will not stop his attack on Ukraine. He wants to have Ukraine under control and he wants a new world order where he will have a major say.
“That’s something miscalculated by the Western nations, that somehow the situation will be resolved for Ukrainian people.
"We’ve seen him attacking Georgia in 2008, we’ve seen him attacking us in 2014 – the war has been ongoing since then, just in a smaller scale."
Asked if the West could have stopped the attack on Ukraine from happening, Ms Klympush-Tsintsadze said: “Yes, I think so.
"If sanctions were taken as pre-emptive measures, that could have been some of the things to deter him. I think you underestimated the craziness of a dictator.”
Ms Klympush-Tsintsadze said the situation could result in another World War. “Putin is not going to stop with Ukraine – if he conquers Ukraine, other nations will be at stake and their future will be at stake,” she said.
President Zelensky has urged Western leaders to encourage anti-war protests in Russia as he said there had been heavy losses on the Ukrainian side as well as heavy losses inflicted on Russia.
His comments come after First Minister Nicola Sturgeon said “this is not a time for half measures” when referring to the sanctions imposed on Russia.
Mr Putin must “feel the wrath of the democratic world” after Russia mounted a “full-scale invasion” of Ukraine, Ms Sturgeon has previously said.
The First Minister said the Russian military action meant Ukraine’s “very existence as an independent democracy was now under attack”.
She made clear she condemns “in the strongest possible terms, the unprovoked, imperialist aggression of Vladimir Putin”.
Olga Stefanishyna, deputy prime minister for European and Euro-Atlantic integration of Ukraine, also urged other world leaders to take action to stop Mr Putin.
She said: “We urge the world leaders to take immediate actions to stop the aggressor: severe economic & financial sanctions, including disconnection from Swift; embargo on oil and gas trade, help army with weapons, UN peacekeeping mission and security guarantees for Ukraine."