Vinicius Junior gave the champions the lead just before halftime with a long-range shot, but the visitors equalised after the break through Kike Garcia, who scored with a brilliant header from the edge of the area that looped over the goalkeeper.
Benzema fired his penalty off the bar and over in the 79th minute, after being fouled inside the area by defender Unai Garcia, who was shown a straight red card after a VAR check.
"We deserved the win but we missed a penalty. It's normal, setbacks happen. This is football," Madrid manager Carlo Ancelotti told reporters. "Benzema has done his part, he had the penalty chance after a very good play but missed it. It's an accident that sometimes happens. We have to move forward."
Osasuna, however, held on to take a point to the frustration of the home fans.
Real Madrid are level with rivals Barcelona on 19 points at the top of standings, with Barca ahead on goal difference.
The point at Santiago Bernabeu lifted Osasuna to sixth on 13 points.
Ancelotti had a major selection headache after Croatia midfielder Luka Modric reported back from international duty nursing an injury and goalkeeper Thibaut Courtois was a surprise late omission after suffering a nerve problem in his back.
"Courtois has a medical test tomorrow and we'll see, it isn't a muscular problem, it's a sciatic nerve problem, we'll see tomorrow," Ancelotti said. "It isn't too serious. We'll evaluate it to see if he can play on Wednesday.
"There are no excuses with the international break, we have to deal with it, the squad is very good. I made some changes because of that, picking players who stayed here. The game didn't turn out the way we wanted, but we could have won, we just didn't have the luck to score the penalty."
On the positive side, Benzema was lively up-front alongside Vinicius in his long-awaited comeback and almost scored early in the first-half from a cross by the Brazil international, firing a splendid first-time volley that went narrowly wide of the left post.
But it was Vinicius who gave Real Madrid the lead with what looked like a lucky effort from distance.
The winger whipped a cross from just outside the left corner of the box towards the back post, but it evaded everyone and somehow ended up going all the way into the net as the goalkeeper took too long to react.
Antonio Rudiger and Benzema were in offside positions and moved for the ball, but the goal stood after a VAR check.
Osasuna found the equaliser five minutes after the break and held on until Garcia fouled Benzema in the six-yard box.
The referee did not initially award the penalty, but changed his mind after checking the VAR footage, showing a red card to the Osasuna defender.
Benzema, however, sent his penalty kick off the crossbar and over.