Two headed goals from Oriol Riera put the hosts in the driving seat before Real's task became even tougher when Sergio Ramos was sent-off before half-time for two bookable offences.
Isco halved the deficit almost immediately after Ramos' departure to give Madrid hope. Despite being in desperate need of a goal, Madrid coach Carlo Ancelotti substituted the in-form Bale just 10 minutes into the second period.
Osasuna were also forced to play the final 12 minutes with 10 men as Francisco Silva was dismissed for a foul on Cristiano Ronaldo and Madrid took advantage straight away as Pepe headed in from the resulting free-kick.
The draw moves Real to within two points of leaders Barcelona and Atletico Madrid, but the Catalans can extend their advantage when they host Villarreal later on Saturday.
Ancelotti hailed the character of his players. "We lost a bit of control of the game after the first goal because we had played very well and with a lot of quality in the first 20 minutes," he said.
"It was incredible to be 2-0 down with 10 men towards the end of the first-half given how we had played. "The team showed great character and attitude to come back and draw the game."
Madrid have now only won once in their last six visits to Pamplona, but having scored 24 goals in their previous five league games, they had started the game confidently.
Ronaldo ought to have given the visitors the lead on 10 minutes when he pulled a shot wide with just Andres Fernandez to beat following a lovely flick from Isco.
However, it was the hosts who went in front with virtually their first attack of the game as Marc Bertran's excellent in-swinging cross was met by the unmarked Riera at the back post.
Madrid were then rightly left aggrieved when Luka Modric appeared to have been hacked down by Alejandro Arribas inside the penalty area, but referee Clos Gomez waved play on.
Six minutes before half-time it was 2-0 as after Diego Lopez had made a fine save from Damia's header, Riera was on hand to nod into an empty net.
Things went from bad to worse for Ancelotti's men moments later when Ramos was harshly sent-off for the 18th time in his Madrid career after clashing with Roberto Torres in midfield.
However, Madrid were given a glimmer of hope before the break when Ronaldo ghosted past three Osasuna defenders and teed up Isco to blast an unerring finish into the bottom corner from 20 yards.
With his side failing to make much of an impression after the break, Ancelotti surprisingly withdrew Bale with over half an hour still to play as Angel di Maria replaced the Welshman.
Lopez made a smart save low to his right to prevent Torres from sealing the points for Osasuna 20 minutes from time.
The numbers were then evened up 12 minutes from time when Silva dived in on Ronaldo and was rightly shown his second yellow card of the afternoon.
And Madrid were on level terms from the resulting free-kick as Pepe climbed highest to power Isco's cross into the bottom corner.