Mohamed Salah scored the only goal of the game three minutes from time as the Swiss champions leapfrogged Schalke into second place behind the Londoners.
The German club's draw with Steaua Bucharest ensured the Premier League club's place in the last 16, while Basel travel to Gelsenkirchen next month knowing a point will see them through.
Basel had ended Chelsea's 29-match unbeaten home record in the Champions League group stages with a 2-1 win at Stamford Bridge in September and they repeated that feat to raise the roof at St Jakob Park.
Chelsea failed to register a shot on target in a lacklustre display, but manager Mourinho refused to blame his players and felt fatigue played a major part.
The Portuguese went on to say he regretted not making more than one change following the victory over West Ham on Saturday.
"It was a deserved defeat, but because of other results we qualified. That is the only positive thing, now we have to get a result at home (Steaua) be first," the former Real Madrid told Sky Sports.
"I didn't like it from the first minute, when there was a defensive mistake immediately. The team only had a little bit of stability and control in the second half.
"The team was tired, they paid the price of the international 15 days. The arrival, the match that followed was crucial against West Ham, where the team played very well in an intense game, but today we paid the price because the players were tired.
"I'm not so sad with them, because I understand. The team was not fresh and maybe I should have made more changes, but there was a temptation to keep the same team because they played so well (at West Ham)."
Striker Samuel Eto'o was carried off just before half-time after falling awkwardly, with Mourinho revealing that the extent of the injury was not yet clear.
