Uchida struck first midway through the second half, after his looped cross found its way into the net, before Boateng slotted home with a calm finish.
Draxler added a thrid with a well-taken chip five minutes from time, to give the German side an early initiative in Group E.
Schalke named an unchanged side from the one that beat Mainz at the weekend, with Boateng handed his first Champions League start for the club, though Jermaine Jones failed to prove his fitness.
Steaua, the only team from Romania in this year's group stage, had the luxury of having virtually a full-strength squad to choose from, with striker Federico Piovaccari making way for Pantelis Kapetanos up front.
The home side dominated possession in the early exchanges, with Draxler finding space on the left flank on several occasions and it took just nine minutes before Steaua goalkeeper Ciprian Tatarusanu was called into action, getting down well to make a smart save from Adam Szalai's volley.
Tatarusanu was on hand again four minutes later, this time doing well to parry away a Boateng header.
The visitors continued to be deprived of possession as the first half wore on, but remained resolute at the back as Schalke looked to release the pace of both Draxler and Jefferson Farfan.
Farfan was on hand to set up what should have been the opening goal after half an hour, playing a low ball in from the right for Boateng, who could only drag his shot wide from 10 yards out.
The hosts were nearly made to pay for their wastefulness around a minute later, when Cristian Tanase stung the palms of Timo Hildebrand in the Schalke goal from distance, in what proved to be a rare first-half attack for Steaua.
Schalke continued to threaten and were unlucky not be ahead seconds before the break, when a Dennis Aogo corner found its way through to Roman Neustadter, whose shot was deflected just wide.
Steaua looked bright after the break, carving out several opportunities with in the first 10 minutes of the half, with Alexandru Bourceanu causing problems for the home defence.
First a whipped Bourceanu corner found Kapetanos, who fired over, shortly before the winger decided to have a go himself, curling a long-range shot just wide.
Another Bourceanu corner caused yet more problems for the Schalke defence just a few minutes later, with the hosts having Uchida to thank after he headed a deflected ball off the line.
And Uchida's night got even better just 10 minutes later, when the Japan international's deep cross eluded everyone before sailing into the net.
The goal brought a sense of relief among the home fans, and 11 minutes later Schalke doubled their lead, when a perfectly-weighted ball from Draxler found Boateng in the area, who applied a calm finish to score.
Draxler himself got on the scoresheet three minutes later, latching on to a threaded through-ball from substitute Christian Fuchs before chipping over the goalkeeper.
It should have been 4-0 as the game advanced into stoppage-time, but Christian Clemens was denied by the woodwork, on what was a great night for Jens Keller's team.