The double blow arrived right at the end of the first half as the Swiss side displayed an immaculate sense of timing to seize the initiative at the halfway stage of this quarter-final meeting.
Boro boss Steve McClaren was denied the services of skipper Gareth Southgate and midfield anchormen George Boateng and Lee Cattermole through injury and suspension and he named a defensive looking line-up in the hope of nullifying Basel on their own ground.
Stewart Downing, Gaizka Mendieta, Ray Parlour and Doriva all earned recalls as Boro looked to keep things tight ahead of the return leg next week.
The spoiling tactic looked like paying dividends for the first 40 minutes of the tie as Boro held firm at the back in the face of heavy Basel pressure.
They even threatened themselves as Mendieta went close with a wicked dipping strike from long range in the 33rd minute which Pascal Zuberbuhler only just managed to tip over the bar.
But any hopes Middlesbrough had of keeping a clean sheet were blown away in two costly minutes late in the first period.
First Matias Delgado crashed in a hopeful 30-yard shot which took a kick off the uneven surface before bouncing over the diving Mark Schwarzer.
Then with the visitors still reeling from the opening blow, David Degen made it 2-0 on the stroke of half-time as he picked up a lay-off from Eduardo and tucked the ball past the advancing Schwarzer.
After the break Boro had plenty of possession as they went in search of the away goal which would have changed the whole complexion of the tie.
First Jimmy Floyd Hasselbaink made the keeper scramble across his line to palm away a deft header in the 50th minute and then Chris Riggott rattled the crossbar with a towering header from a Mendieta corner.
At the other end the lively Delgado hit the woodwork with a free-kick from an acute angle before Riggott saved his side with a goalline clearance from Scott Chipperfield's shot.
To Boro's credit they refused to give up and substitute Ugo Ehiogu was inches away from forcing a header over the line in the 89th minute.
But after drawing a blank away from home, Middlesbrough will have it all do back at the Riverside if they are to reach the last four.