Wayne Rooney struck twice to bring up 200 goals for the club, but it was a controversial opener from the England striker midway through the first half to set United on course for victory in their Group A opener.
It looked like United would be made to pay for a shocking Rooney miss when Leverkusen drew level in the 54th minute as Simon Rolfes' long-range effort deflected wickedly off Michael Carrick.
However, Robin van Persie quickly restored United's lead and Rooney made it 3-1 before Antonio Valencia's excellent goal on the counter-attack in the 79th minute.
Omer Toprak grabbed a late consolation for the German side but it did not spoil a wonderful start to their latest European campaign for the Premier League champions.
Maroaune Fellaini and Shinji Kagawa were brought in for starting roles and United began brightly, as Rooney had an early shot blocked by Emir Spahic, with Van Persie also firing wide in a solid if unspectacular start.
Chris Smalling, making his first appearance of the campaign, looped an effort over off his back before United opened their account in the 22nd minute.
A Ronaldo-esque dance from Van Persie by the touchline bamboozled the Leverkusen defence, then Fellaini, Michael Carrick and Kagawa were all involved in sending Patrice Evra darting towards the by-line. His cross was met by Rooney, whose shot bounced over goalkeeper Bernd Leno.
What Slovenian referee Damir Skomina - and the rest of his officials - had missed was Valencia virtually falling into Leno as Rooney's shot came towards them. If Valencia was not interfering with play, it is hard to see how anyone ever could be.
Van Persie's decision to engage in a face-to-face row with Skomina was not the wisest choice and brought him a booking.
Still, with Kagawa coming agonisingly close to a second just before the interval, United did not have a lot to concern them.
That confidence was badly shaken not long after the restart. As he glided round Leno, Rooney looked certain to tap home. Instead, he rolled his effort beyond the far post, too far in front of Van Persie to enable the touch that would also have doubled United's lead.
Both Moyes and, in the directors' box, Sir Alex Ferguson, struggled to comprehend how United were not two up.
To make matters worse, the visitors immediately went down to the other end and equalised as Leverkusen skipper Rofles met Heung-min Son's lay-off perfectly and saw his curling effort take a deflection off slight Carrick before finding the corner.
It took a moment of genius from Van Persie to restore United's advantage, hooking Valencia's cross in with a semi overhead kick that left Leno so startled he could not react quickly enough.
Once they were back in front, United were in no mood to surrender the initiative again.
Rooney's landmark goal came 20 minutes from time as he took advantage of poor defending from a long David de Gea pass to lift a shot neatly past the onrushing Leno and it at the near post.
Rooney then set up Valencia for his first European goal in nearly two years following a sweeping counter-attack also involving substitute Ashley Young.
Toprak pulled one back after a goalmouth scramble caused by De Gea's failure to punch clear before Van Persie failed to convert from barely a yard.
Still, for Rooney - who left to a standing ovation - and Moyes, it was a very pleasing evening indeed.