The Colombian put Bayern ahead in the fourth minute at the Parc des Princes and took advantage of the weak defense to score again shortly after the half hour mark.
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However, the former Liverpool winger was red carded in first-half stoppage time after a rash challenge on Hakimi forced the Moroccan PSG star to leave in tears.
Joao Neves scored for PSG, but Bayern held on despite playing the entire second half a man down and despite Harry Kane failing to add more than his 25 goals for club and country this season.
Bayern’s sensational start to the campaign has seen them win all 16 games in all competitions, including four in the Champions League: they are one of only two clubs with 12 points in Europe’s elite competition alongside Arsenal.
“I know that the winner of the Champions League is not decided now, otherwise PSG would not have won it last season,” said Bayern coach Vincent Kompany.
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“But now it’s just about getting to the next round and these were three very important points.”
As Bayern move forward, it remains to be seen what the consequences will be for PSG of the injury to Hakimi, who must lead Morocco to the African Cup of Nations at home next month.
PSG also saw Ballon d’Or winner Ousmane Dembélé come off in the first half, interrupting just his second start in two months after a hamstring injury.
“We have to be attentive and try to improve the physical condition of the players, but it is not easy considering how many games we have left to play,” PSG coach Luis Enrique told Canal Plus.
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His team, which won last season’s Champions League final in style against Inter Milan in Munich, is now plagued by physical problems: Desire Doue, the best player in that final, was already absent here due to injury.
However, the actual consequences of losing this match may be less problematic. PSG reached the top of the Champions League standings this week and remains well positioned to qualify for the knockout rounds.
“I can’t talk about the future. What we do know is that we have the most difficult games,” said Luis Enrique, as PSG faces Tottenham Hotspur.
This was a heavyweight encounter between the Champions League’s top two after the first three games, and a repeat of the 2020 final, won 1-0 by Bayern in Lisbon.
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-Hakimi cries-
It was also a repeat of the Club World Cup quarter-final won 2-0 by PSG in Atlanta in July, and remains the German giants’ last competitive defeat.
This was a huge test of their credentials but the visitors scored from their first start as Michael Olise was denied by goalkeeper Lucas Chevalier only for the loose ball to fall to Diaz to shoot.
The hosts had created nothing before momentarily thinking they were tied midway through the first half when Dembélé chested down Fabián Ruiz’s shot.
However, the effort was disallowed for offside and Dembélé was replaced, making way for Lee Kang-in and heading straight towards the tunnel in what PSG fans will hope was a precautionary measure.
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A flurry around the half-hour mark saw Bradley Barcola approach for Paris and Serge Gnabry attempted a shot at the other end which hit one post and grazed the other on its way out.
Bayern then scored again in the 32nd minute when Díaz robbed Marquinhos just outside the box and scored.
A Josip Stanisic goal for Bayern was ruled out before Diaz went from hero to villain on the stroke of half-time when he clumsily brought down Hakimi, with the Moroccan’s left ankle twisting in the challenge.
As Hakimi sobbed and was helped out, the referee gave Diaz a yellow card and then upgraded it to a red after a review.
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PSG struggled to make the extra man count in the second half before substitute Neves fired in from close range following Lee’s deflected cross in the 74th minute. That set up a soapbox finish, but Bayern held on.
as/bsp