Valencia reeled in Basel's 3-0 first-leg lead to force extra-time in their quarter-final second leg on Thursday and Yakin's side fell apart in the extra half-hour, as Diaz and Sauro were sent off before the Spanish hosts scored twice to win 5-3 on aggregate.
While Yakin said Valencia were better 'in all respects', he was furious with how Diaz and Suaro got themselves red-carded.
Diaz kicked out at Jonas 10 minutes into the extra period, while Sauro earned two yellow cards in the space of a minute to be sent off just before the break in extra-time.
"Such situations need to be addressed when players let out their frustration on their own insufficient power, in actions that do not benefit the team," Yakin said after the match.
Basel were very confident of reaching the Europa League semi-finals before Thursday's match, after they won the first leg comfortably, but when Paco Alcacer and Eduardo Vargas scored just before half-time in the second leg, Valencia turned the contest around.
"The two goals before the break was too much," Yakin said.
"We must not delude ourselves, Valencia was in all respects - especially mentally – better."
Basel goalkeeper Yann Sommer agreed: "After a quarter of an hour, I realised that it could be really difficult and after the 0-1, which was rather stupid, their hope returned.
"The 2-0 was for them the crucial scene in the game and broken our neck. We continued to fight and then the two red cards came. (Losing) 0-3 I can accept but not 0-5."
The match was Sommer's last for Basel at the continental level before he leaves for Borussia Monchengladbach at the end of the season.
While the 25-year-old goalkeeper was disappointed that Basel's Europa League campaign was over, he maintained the Swiss Super League title remained their priority, as they chase a fifth straight championship.
"The Europa League is a bonus for us and yet we are now very disappointed," Sommer said.
"But this must be so. Already on Friday, we will look forward to Sion and the championship.
"The important part of our football experience is the championship and we want to finish well."
Basel lead the Swiss Super League by one point from Grasshopper with eight games remaining.
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