Diaz scored once in each of three group matches, and his 64th-minute strike against the Tanzanians created history as he became the first Moroccan to find the net in four consecutive Afcon matches.
His celebrations included holding up a shirt belonging to midfielder Azzedine Ounahi, and he later dedicated the last-16 victory to his injured teammate.
Ounahi arrived at the stadium using crutches and wearing a medical boot on his left foot as Bilal El Khannouss took his place in midfield.
The goal came after Morocco squandered numerous scoring chances while stretching an unbeaten run in competitive and friendly matches to 23. Their last loss was to South Africa at the 2024 Afcon.
Morocco will face South Africa or Cameroon, who meet later on Sunday at a different Rabat venue, in the quarter-finals on Friday.
Paris St-Germain defender Achraf Hakimi set up the last-16 winner for Diaz while making his first start in the tournament after coming off the bench in a victory over Zambia last Monday.
The 2025 African Player of the Year suffered a serious ankle injury playing for PSG against Bayern Munich in the Champions League two months ago.
“The competition is growing in intensity, and we have just faced our toughest opposition so far in Tanzania,” said Diaz.
“Not everything worked, we know that, but fortunately we managed to secure our qualification. Now we will go back to work to be fully ready for the quarter-finals.”
Argentina-born Tanzania coach Miguel Gamondi said, “I am so proud of what my team produced. So proud of my players, both in their attitude and tactically.
“They showed that the gap between Morocco and us is not as big as people think. We will come back stronger, I am convinced of that.”
First chance
Morocco were favourites for several reasons, including home advantage, the backing of close to 70,000 supporters, and lying 101 places above Tanzania in the world rankings.
However, it was Tanzania who had the first chance in the third minute, but Saimon Msuva misconnected with a Selemani Mwalimu cross.
Morocco had the ball in the net after 15 minutes as Ismael Saibari nodded an Abdessamad Ezzalzouli free kick past goalkeeper Hussein Masaranga.
The assistant referee immediately raised his flag for offside, and a VAR review confirmed that the PSV striker had strayed too far forward.
Ayoub El Kaabi was the next Moroccan to come close, heading over before being injured in a collision with Masaranga. Both players resumed after treatment.
A Diaz shot was too high from the edge of the box, and El Kaabi headed wide as the host nation continued their pursuit of the opening goal, but the first half ended goalless.
In a lively start to the second period, Ezzalzouli and El Kaabi headed wide, then, in a rare Tanzanian raid, Feisal ‘Fei Toto’ Salum fired over with only goalkeeper Yassine ‘Bono’ Bounou to beat.
As Moroccan pressure mounted, Hakimi rifled a free-kick against the crossbar from just outside the box on the hour mark.
A goal seemed inevitable given the relentless pressure from the host nation, and it came thanks to Hakimi and Diaz.
The full-back passed to the forward, who beat Masaranga at his near post with an angled shot from close range.
Tanzania protested furiously in added time after El Khannouss appeared to push Tanzania substitute Iddy Nado in the area, but the referee waved play on.