The much-coveted midfielder scored a sensational goal shortly after half-time to maintain the German side's 100 per cent start in Group E
Marco Streller wasted several good opportunities for the hosts and Max Meyer hit the crossbar before Draxler lashed home from the edge of the area just after half-time in a game that was delayed early in the first half due to a Greenpeace protest against Gazprom, sponsors of the competition and of the German side.
The win means leaves Jens Keller's side top of the group, having not lost away in Europe in 12 games.
Basel's Fabian Schar managed to overcome a knee problem in time to receive a place in the starting line-up, while Taulant Xhaka was preferred ahead of Giovanni Sio.
Klaas-Jan Huntelaar, Kyriakos Papadopoulos, Christian Fuchs and Jermaine Jones, were all sidelined for the visitors, although Draxler managed to shake off a knee injury.
Both sides enjoyed a winning start to their Champions League campaigns and came into this contest knowing that another victory would strengthen their position in Group E.
Everyone in the stadium in Basel were left bemused, however, when the game was halted within the first few minutes after Greenpeace activists abseiled from the roof to unveil a banner criticising Gazprom.
When the action restarted after a break of just over five minutes, Schalke soon set about carving out the first opportunity, with Kevin-Prince Boateng firing wide after just over 10 minutes.
The disruption to the game clearly impacted on the rhythm of both sides, with chances few and far between during the opening exchanges.
Basel's best opportunity of the first period came just after the half-hour mark when Streller controlled in the box before attempting an acrobatic effort that forced Timo Hildebrand into a smart save.
Murat Yakin's Basel side began to settle, with Mohamed Salah looking particularly lively, as the hosts continued to look for a breakthrough.
Schalke were dealt a blow just before half-time when winger Jefferson Farfan limped off to be replaced by Adam Szalai.
Streller then headed a whipped Kay Voser cross just past the post.
The visitors responded well after the break and were unlucky not to be ahead when Draxler's teasing cross was only headed as far as Meyer, who sent a crashing strike back off the bar.
Draxler was at the centre of proceedings when Schalke took the lead just a couple of minutes later. A Dennis Aogo corner found its way to the 20-year-old on the edge of the box, who rifled a sweetly-struck half-volley past a despairing Yann Sommer.
Schalke predictably began to look confident in the middle of the park, although Salah continued to pose a threat to the visitors' defence.
Salah was on hand 15 minutes from time to force Hildebrand to palm a stinging effort around the post.
Yakin threw on Matias Delgado and David Degen in an attempt to unlock the Schalke defence, but the hosts continued to threaten.
Draxler nearly doubled his tally late on, but could only fire over after a good run down the left-hand side, as Keller's side held onto a well-earned win.