"There was some warning of a possible attack and police were monitoring, but we did not know where it would happen," Phrewphan Damapong told reporters.
The city has been on edge since Thai police charged a Lebanese man suspected of planning an attack in Bangkok following a US warning of a threat of a terrorist strike on tourist areas last month, although Thai police said they had found no direct link.
A statement by Israel Defence Minister Ehud Barak suggested the incident was part of a wider threat.
"The attempted attack in Bangkok proves once again that Iran and its proxies are continuing to act in the ways of terror and the latest attacks are an example of that," said the message released in Hebrew.
Tuesday's blasts came a day after bombs targeted Israeli interests in India and Georgia, injuring a diplomat and her driver in New Delhi, in attacks that Israel accused Iran of masterminding.
Bangkok bomb squad investigators rushed to the scene after three men were seen fleeing after a blast at a house in the Sukhumvit Road area in the east of Bangkok.
One of the men hurled an explosive device at a taxi which refused to stop and later tried to throw another device at police, triggering a blast on the side of a busy road which tore off his legs, authorities said.