The striker capitalised on poor defending at a corner to beat Petr Cech at his near post and seal the win for the Swiss side, who had trailed to Oscar's first-half opener before Mohamed Salah levelled with 19 minutes remaining.
Chelsea had not previously lost a Champions League group game in almost 10 years, and a triumphant Streller told Sky Sports: "We always expect it, but we knew it would be a tough game.
"After [Basel got back to] 1-1 I felt Chelsea had a problem and I said to my boys 'listen, we can win here' … and they listened to me! I want to say it's a miracle. Now we are getting more experienced and we can win here in Chelsea.
"It's unbelievable. I think people in England will know that we are a team to be reckoned with."
Basel coach Murat Yakin made it clear that much work still lies ahead for his side if they are to qualify from Group E, which also contains Schalke and Steaua Bucharest.
"We will remain humble now and know that all we did was win the first game," said Yakin.
"[I owe] A huge compliment to my team. After years we have finally managed to win a game over here. We will continue to focus on one game at a time."