The euro dropped below the $1.16 line for the first time in 6-1/2 months, down 0.3 percent on the day. Against the Swiss franc, it fell by a similar margin at 1.1528 francs.
MSCI's broadest index of Asia-Pacific shares outside Japan was 0.1 percent higher, but Japan's Nikkei stock index fell 1.2 percent as auto shares slumped.
Swadeshi Jagran Manch, a wing of Hindu nationalist organisation Rashtriya Swayamsevak Sangh (RSS), protested on the roads of New Delhi, shouting slogans such as "Go back Walmart", reminiscent of "Go back Simon", a popular anti-British slogan during the country's freedom struggle.
Lagarde's advice came as the US Treasury Secretary Steven Mnuchin said on Saturday he was considering a trip to Beijing to discuss trade issues with his Chinese counterparts.