The 19-year-old striker has made a habit of netting in his maiden appearances.
With United cruising two goals to the good against the Swiss side, Rashford swooped to put the clash beyond any doubt and make it 3-0 to Jose Mourinho's men.
It took him just seven minutes to beat the Basel defence after coming on as a late replacement for Juan Mata in the 77th minute.
In putting Basel to the sword, Rashford also followed in the footsteps of Wayne Rooney in scoring in a Champions League fixture as a teenager.
Only Rooney, David Beckham and Phil Jones had made the scoresheet for United before their 20th birthday, and now the young striker joins the club.
The sky really seems the limit for this wonderkid, vindicating Mourinho's decision to bring him off the bench in his first Champions League outing as United boss.