The Trump administration's decision to restore sanctions "is the sea change the Middle East has been waiting for", he said.
"That's the goal, that's the mission, and that's what we will achieve on behalf of the president".
"Today, Iran is able to sell its oil and it will sell", Rouhani vowed Monday as the sanctions kicked in.
Pompeo said earlier on CBS' "Face the Nation" that he was "confident" Iran would not restart its nuclear program with the US withdrawing from the deal.
"Watch as we have already taken more crude oil off the market than any time in the previous history", he added, referring to the one million barrel reduction in Iranian oil exports already.
Wallace moved to another point, asking Pompeo if the sanctions are enough, citing the fact that Senators Ted Cruz (R-TX), Tom Cotton (R-AR) and Marco Rubio (R-FL) "plan to introduce legislation to cut off Iranian banks from the global financial network known as Swift".
Israel is particularly concerned with Iran's involvement in neighbouring Syria and has pledged to keep it from entrenching itself militarily there.
He described the sanctions as the "strongest tool" the U.S. holds, before resorting to military means, to pressure the Iranian regime to adhere to its demands. "But they're not doing very well", he said.
"All eyes will be on Iranian exports, whether there will be some cheating around U.S. sanctions, and on how quickly production will fall", said Riccardo Fabiani, an analyst for Energy Aspects.
The UK, Germany and France - which are among the five countries still committed to the nuclear pact - have all objected to the sanctions.
Oil-buying nations, including China, India, Japan, Turkey and South Korea, will be given permission by the Trump administration to purchase Iranian output, although no European countries are included in the list, according to reports.
In the capital city of Tehran, the rally was held at the venue of the former USA embassy, which is known by Iranians as the "Den of Espionage".
At a gathering on Saturday, Iran's supreme leader, Ayatollah Ali Khamenei, portrayed the sanctions as part of a long tradition of American hostility toward the Islamic Republic, which he said had nevertheless prevailed.
What has the reaction been in Iran?
Meanwhile, in Tehran, demonstrators poured onto the streets to mark the 39 anniversary of the occupation of the US embassy there, an action that precipitated a protracted hostage standoff that helped tank President Carter's chances at re-election and almost took down the Reagan administration amid the Iran-Contra scandal.
Hardliners hold protests to commemorate the siege every year but on Sunday, protestors also vented their fury about the sanctions. The government sources said the finance ministry has given its nod to Iranian bank Pasargad to set up a branch in Mumbai, which will also help India to continue its oil supplies with Iran and make payment from the country. We are confronting a bullying enemy. "If prices start to rise again or another major producer has difficulties, it could put pressure on the United States and lead to new exemptions", said Fabiani.