Ramos had won the trophy twice before at Sevilla and a third triumph would put him level with Republic of Ireland boss Giovanni Trapattoni.
However, a first half goal from Valentin Stocker and a second half strike from Marco Streller at the St. Jakob-Park ensured the Swiss side head into next week's second leg at the Dnipro Stadium in control.
Basel - who are second in their domestic competition - qualified for the last 32 after victories over Videoton then Sporting Lisbon saw them qualify after they failed to win their first three encounters.
Dnipro's passage was far more comfortable, with victories over Napoli and PSV seeing them into the knockout stages.
And Ramos' side almost went a head just 16 seconds into the contest when Nikola Kalinic's shot from inside the area had to be tipped round the post by home stopper Yann Sommer.
His opposite number Jan Lastuvka was called into action for the first time after 11 minutes as he kept out Stocker's 20-yard piledriver.
However, there was nothing he could do after 22 minutes to deny Stocker. Adilson Cabral headed down for the midfielder who drove powerfully past Lastuvka and into the net.
Desperate to keep a much-needed clean sheet, a block by Fabian Schar denied Ruslan Rotan's goal-bound header, before Yevgen Konoplianka's 20-yard effort was caught by Sommer, who showed good handling six minutes later to keep out Nascimento Matheus' effort.
Serey Die shot straight at Dnipro's keeper after 63 minutes from inside the area when he should have done better, and just two minutes later it was two.
Stocker this time the provider as his near post corner was hit into the bottom left-hand corner by Strellar.
Sommer did well to keep out Ruslan Rotan's free kick late on, but Dnipro will have it all to do in the Ukraine.