The Catalan side could have reduced the gap to three points but were sunk by an Alex Berenguer double and an Inaki Williams goal to remain six adrift of Madrid.
It was another stumble for a side that matched the 35-times champions blow for blow in the first half of the season.
After going more than three months without defeat, Girona have now taken just one point from their last three games - which included a 4-0 hammering at Madrid on February 10.
Berenguer put Athletic ahead in the second minute when he turned the home defence and fired home a left-foot shot from outside the box.
Viktor Tsyhankov equalised early in the second half for Girona, but Berenguer finished superbly seven minutes later to give Athletic the lead again.
Williams made it 3-1 when he took advantage of poor defending and scored with a low shot in off the post.
Eric Garcia headed home Girona's second with 15 minutes remaining, but although they pushed hard in the final minutes the visitors could not level the scores.
Fifth-placed Athletic reduced the gap to two points with Atletico Madrid, who occupy the final automatic Champions League spot.
"Of course we're going to dream about the Champions League," goalscorer Williams said.
"We're taking it step by step, as we've been quietly saying.
"We're working and pushing ourselves to the max so that the success makes the noise."
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