Madrid looked set for a first home league defeat in a year as Las Palmas took advantage of Bale's dismissal early in the second-half to lead 3-1 with just three minutes remaining.
However, Cristiano Ronaldo's double salvaged a draw and keeps Real just a point adrift of league leaders Barcelona with a game in hand.
"We can't look for excuses," said Zidane.
"What we can do is improve our performance. We are not happy with how we are playing.
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Bale's first red card since joining Madrid for a then world record fee in 2013 came with the scores level at 1-1 as Tana cancelled out Isco's opener for Real inside the opening 10 minutes.
The Welshman kicked out at Jonathan Viera twice before finally pushing the Las Palmas midfielder to the ground.
"He has apologised. He is not happy with the sending-off," added Zidane.
Madrid vice-captain Marcelo said Bale is aware his moment of madness had badly let his teammates down.
"Bale is old enough to know if he has made a mistake or not. These things happen in football, but hopefully it doesn't happen again between now and the end of the season."
In a recurring theme of Madrid's season, a stunning late effort undid a lot of the damage caused by a below-par display.
The European champions trailed Villarreal 2-0 on Sunday before a rousing fightback in the final 25 minutes sealed a 3-2 win.
- 'We will change things' -
However, having also lost 2-1 at Valencia last week after conceding twice in the first 10 minutes, Zidane insisted Los Blancos can't always leave themselves with a mountain to climb.
"We can't go over the top. There are things we have done well this year, but we have to analyse it and we will change things," added the Frenchman.
"We have shown we have character and we can do more. We have to improve particularly in our first-half performances. We have to think about the positives. We will analyse the negatives, but to come back from 1-3 to 3-3 is a huge effort."
A 6-1 thrashing of Sporting Gijon earlier on Wednesday saw Barca leapfrog Madrid into top spot and the two sides are still to meet at Real's Santiago Bernabeu in April.
However, Zidane rejected suggestions Madrid are in danger of throwing the title away.
"We are going to fight. When there were good moments and we were leaders, I always said we would have difficult moments. "We are not leaders, but the league is still open and we are not going to give up. "You win leagues at the end (of the season) not now.