Steaua, the 1986 European champions, only had Dica's first-half strike to show for their dominance as keeper Mark Schwarzer kept injury-hit Boro in the tie.
Steaua coach Cosmin Olaroiu said: "Before the match I would have been content with a 1-0 win, but now the victory has a bitter taste."
Australian Schwarzer, who had no chance of saving Dica's well-struck shot on the turn after 30 minutes, made two fine saves in the second half as Steaua pressed strongly.
He tipped over Dorin Goian's volley from close range and minutes later got down well to smother Dica's shot after a defensive lapse.
Boro, who went into their first European semi-final with a re-shuffled defence, enjoyed their best spell late in the first half. Nigerian Yakubu Aiyegbeni's downward header bounced up and hit the bar and then James Morrison forced a save out of Carlos Fernandes.
Steaua's uncompromising style brought four first-half bookings, ruling out midfielders Sorin Paraschiv and Banel Nicolita from the return at the Riverside next week.
Middlesbrough manager Steve McClaren, whose side play an FA Cup semi-final against West Ham United on Sunday, was content to escape with a single-goal defeat.
"The 1-0 defeat is not a bad result," he said. "We are even delighted with it, because it could have been worse.
"At home, it will be another atmosphere and a different match."
Captain Ugo Ehiogu admitted Boro had under-performed but was hopeful of turning the tie around.
""We knew they would make it difficult... these are regular European campaigners," he said.
"They came out more than we expected but on the whole, a one goal deficit... we know we can score goals at home.
"We had to dig in at times. It was difficult, we were carrying a few injuries but that is no excuse. Hopefully we'll get some players back we'll give them a better game next week."
Schalke, the 1997 winners, could find no way to breakdown the resilient Spaniards at the AufSchalke Arena.
Sevilla, like Middlesbrough making a first appearance in the last four of a European competition, were grateful to keeper Andres Palop who made a series of fine saves.
Schalke's Serbian defender Mladen Krstajic was sent off for elbowing Jesus Navas 10 minutes from time.
"We can't really be satisfied with the 0-0 result but we've been strong most of the season in away matches," said Schalke coach Mirko Slomka. "Everything is still wide open for the second leg.
Slomka's counterpart Juande Ramos said the Bundesliga side still had the edge despite the Spaniards having home advantage for the second leg.
"The 0-0 draw was about the best we could come away with here," he said. "We've got a good chance at home, although I would say at this point Schalke has a slight advantage (because of the away goal rule)."