Basel required a point at Schalke on Wednesday to reach the knockout phase, but a 2-0 loss resigned them to UEFA Europa League football.
Severely hindering the Swiss side's hopes was the 31st-minute dismissal of defender Ivanov for bringing down striker Adam Szalai. Schalke duly capitalised on their numerical advantage, with goals from Julian Draxler and Joel Matip within 15 minutes of the restart sending them through as Group E runners-up behind Chelsea.
Yakin was pleased with the way his side started the game in Gelsenkirchen against a side who reached the Champions League semi-finals in 2010-11.
"We opened the game well," said the 39-year-old former centre-back, who played in Germany for Stuttgart and Kaiserslautern.
"Our passing was good, we kept calm and had a lot of possession. Unfortunately we then had a player sent off and this caused an imbalance in the team.
"Schalke then scored a deserved lead, but I still felt that we would be able to recover from going down one goal. But then they scored the second goal and the game was over."
Basel began their Group E campaign with a 2-1 win away to Chelsea and also beat the London club at home.
However, back-to-back draws with Steaua Bucharest and two defeats to Schalke ultimately left Basel with eight points from six games, two behind the second-placed Germans.
"Congratulations to Schalke for reaching the next round," continued Yakin.
"At least we will play on an international level next year. But obviously it's a huge disappointment that we failed to reach the next round in the Champions League.
"That's also the reason that I didn't allow myself to be euphoric after we won against Chelsea. I know that in football you have to play all the games before you can celebrate."
Basel will return to the continental stage in the Europa League's round of 32 in February. In the same competition last season, the Swiss side eliminated Zenit St Petersburg and Tottenham en route to the semi-finals, before losing to eventual champions Chelsea.
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