Jose Mourinho's side appeared lost without inspirational top scorer Cristiano Ronaldo, suspended for the game in Pamplona, and unbeaten leaders Barcelona can go 18 points clear of their arch rivals with a win at fourth-placed Malaga on Sunday.
Real stay third on 37 points, four behind Atletico Madrid, who meet Real Zaragoza at home on Sunday.
In a poor-quality and bad-tempered clash at Osasuna's Reyno de Navarra stadium, the main incident came in the 76th minute when Kaka was shown a second yellow card for preventing Osasuna taking a quick free kick.
The little-used Brazilian playmaker had been on the pitch for a little over a quarter of an hour and had been booked moments after coming off the bench for jumping with a flailing arm.
It was the 2007 World Player of the Year's first sending off since he joined Real from AC Milan in 2009 and his first red card since he was dismissed playing for Brazil against Ivory Coast at the 2010 World Cup.
Osasuna's well-earned point lifted them off the bottom of the table above Deportivo La Coruna, who play at Real Sociedad on Sunday.
(Reporting by Iain Rogers, editing by Stephen Wood)