The Romanian media had expected nothing less than a Steaua victory beforehand but Sheriff produced a plucky performance in their first-ever match in the group stages of a European competition.
The visitors looked the more determined side for much of the Group H game and might have won had Robinson Zapata not produced a fine save from Alexey Kuchuk early in the second half.
The game itself offered little excitement, particularly in the first half after two early Steaua opportunities.
Romeo Surdu shot wide from 12 yards after three minutes and Sorin Ghionea then forced a save from Stanislav Namasco with an effort from the edge of the area.
But it was the Moldovans who then had the better of the possession even though they were unable to carve out clear-cut chances.
One good move almost put Kuchuk in on goal but Ghionea just did enough to put him off and prevent the shot.
It was not until just before half-time when Sheriff, for all their industry, were able to work Zapata, but the home goalkeeper kept out Luis Antonio Rodriguez's long-range free-kick.
Kuchuk then almost stole in on the stroke of half-time but fired over.
Those efforts encouraged Sheriff and they were denied the lead early in the second period only by a brilliant point-blank save from Zapata.
Sergio Da Costa broke clear on the right and delivered a superb cross for Kuchuk but Zapata somehow scrambled back to claw his volley off the line.
That woke Steaua up but Namasco dealt with an effort from Emil Ninu and then tipped a long-range shot from Andrei Ionescu round the post.
Time then began to run out for Steaua and Pantelis Kapetanos was unable to find the target from the edge of the box with 11 minutes remaining.
The visitors successfully managed to hold out and could have snatched victory at the end when Kuchuk's cross picked out Vitalii Bulat but his header went over.