Hajsafi's swift control and scything strike from the corner of the box raised the roof at Melbourne Rectangular Stadium as Team Melli opened with a pulsating 2-0 win.
Earlier, UAE's top-rated midfielder Oman Abdulrahman bewitched Gulf rivals Qatar in a 4-1 rout achieved through braces to both Ahmed Khalil and Ali Mabkhout in Canberra.
The UAE went behind to "Qatari Maradona" Khalfan Ibrahim's sweetly executed volley before they roared back to put a huge hole in the campaign of the 2022 World Cup hosts.
The atmosphere in Melbourne was at fever pitch from the start and Iran almost opened the scoring when Ashkan Dejagah forced a smart stop from Sayed Abbas in the Bahrain goal.
Bahrain's Nigerian-born striker Jaycee Okwunwanne blazed just wide and Sayed Shubbar headed a golden opportunity over as Bahrain threatened to capitalise on Iran's sluggish start.
But with nearly the last kick of the first half, Hajsafi struck when he juggled a clearance from his left to his right foot and smashed it into the top corner.
Okwunwanne knocked Iran 'keeper Alireza Haghighi backwards with a ferocious drive before Masoud Shojaei popped up to steer home a mishit effort from a corner.
"The first game is always difficult. After 10 minutes we started to settle and control the game and the only thing is I believed we should have scored one more goal," said Iran coach Carlos Queiroz.
"We deserved one more goal. But to play against Bahrain is difficult."
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