Having taken over from the sacked Julen Lopetegui following the 5-1 Clasico drubbing at the hands of Barcelona, Solari showed early promise with four wins from four in his interim tenure.
That run was enough to earn the former Madrid midfielder a deal through to June 2021, but his fortunes will have to improve drastically if he is to have any hope of remaining in the Santiago Bernabeu hotseat until then.
Gonzalo Escalante's scrappy opener saw Madrid fall behind for the first time under Solari, and the writing was on the wall with quickfire goals early in the second period through Sergi Enrich and Kike.
The shocking defeat capped a tough 48 hours for the club, who on Friday released a statement to deny accusations made by Der Spiegel that captain Sergio Ramos had broken anti-doping regulations on two occasions in the past.
Ramos and the rest of the backline were soon under pressure, Thibaut Courtois producing a wonder-save to tip Kike's dipping 30-yard volley onto the post, before Karim Benzema was denied on the line by Cote.
After 15 minutes of frenetic action, Eibar broke the deadlock as Barca loanee Marc Cucurella surged down the left and squared for Kike. His attempt was brilliantly kept out by Courtois but Escalante managed to bundle the ball home from inside the six-yard box.
And shortly after the restart, the European champions' woe worsened.
Toni Kroos played a poor pass across the defence to Alvaro Odriozola, who dallied on the ball, allowing Cucurella to steal in and tee up Enrich for a low finish inside the left-hand post.
Five minutes later, the game was effectively put beyond a desperate Madrid as Cucurella whipped in a low cross that took a touch off Enrich and ran perfectly for Kike to tap into an empty net from five yards to seal Eibar's first ever league win over their illustrious opponents.
Solari under the pump immediately
Things could hardly have gone worse for the new head coach. One defeat in isolation is far from disastrous, but the manner of such a loss was concerning. Leaders Barca could be seven points clear of Madrid by the end of Saturday.
Madrid malaise returns with a vengeance
Often under Lopetegui, Madrid looked devoid of inspiration. In Solari's four interim games in charge, there were signs of improvement, but this capitulation was a major step back. The Bernabeu board may be wondering if they showed faith in the right man.
