A cagey first period looked set to end goalless until Sikan broke the deadlock in the 41st minute, guiding a header beyond Marc-Andre ter Stegen to register his third European goal of the season.
After soaking up the Barcelona pressure in the second period, Shakhtar thought they had doubled their advantage in the closing stage when Newerton curled a sumptuous strike into the far corner.
The Brazilian's effort was ultimately chalked off for an offside infringement, but it did little to dampen Shakhtar's mood as they held firm to claim a memorable victory.
As a result of their second win in the group stage, third-placed Shakhtar move to within three points of Barcelona, who currently sit at the top of Group H.
After claiming a 2-1 victory when the two sides met on matchday three, Barcelona would have been confident about their chances of picking up another victory in the reverse fixture.
While the La Liga champions controlled possession in the early stages, a determined Shakhtar side were seemingly keen to avenge last month's defeat.
The designated hosts went close to opening the scoring in the 15th minute when Mykola Matviyenko latched onto a long ball over the top before forcing Ter Stegen into action.
Although Raphinha tried his luck with a wayward long-range effort, Barcelona failed to move out of first gear in the opening 45 minutes.
With the visitors unable to hit their straps, Marino Pusic's side took full advantage, carving out a well-worked goal in the final moments of the first period.
After seeing his side enjoy a spell of possession, Georgiy Sudakov chose the opportune moment to switch the play to Giorgi Gocholeishvili, who delivered a ball into the box for Sikan to steer home with a precise header.
Gocholeishvili nearly added his own name to the scoresheet after the break, but he could find no way past the resolute Ter Stegen.
The Barcelona goalkeeper was required again before the hour-mark, standing firm to deny Sikan his second goal of the evening.
Xavi's troop attempted to pin their opponents back as the contest wore on, but they were finding clear-cut chances hard to come by against a well-drilled Shakhtar defence.
The Ukrainian outfit briefly thought Newerton had doubled the advantage in the 87th minute after the Brazilian cut inside before curling an unstoppable strike past Ter Stegen.
Unfortunately for Newerton, he was denied his first Shakhtar goal by the assistant's flag, but it proved inconsequential for the result, as the designated home side stood firm to condemn Barcelona to their first European defeat of the season.