Sami Hyppia's men went into the game knowing they needed to better the Ukrainian champions' result and they deservedly went ahead four minutes into the second half when defender Omer Toprak slammed home from close range.
Leverkusen should have added to their advantage as Stefan Kiessling hit the post and was also denied by a fine save from Enaut Zubikarai.
And the Bundesliga side were nearly made to pay for their profligacy as Antoine Griezmann forced Bernd Leno into an important save with a low header 17 minutes from time.
United's victory means they top the group, while Leverkusen are likely to face a tricky tie against another group winner when the draw for the last 16 is made in Nyon, Switzerland on 16 December.
Real Madrid put the seal on a fine Champions League group stage campaign as they won 2-0 at FC Copenhagen - even if Cristiano Ronaldo did miss a penalty in the process.
Carlo Ancelotti's men have topped Group B by some distance and finish with 16 points from 18 after a record-breaking night.
Luka Modric's fine opener was the 31st European game in which they have scored, setting a new competition record, before Ronaldo made it safe after the break.
The Ballon d'Or nominee had a chance to make it two late on from the spot but those in the Danish capital were stunned as the normally-faultless frontman's effort was saved.
The loss, although expected, leaves Copenhagen out of both the Champions League and UEFA Europa League.
The match was halted shortly after the half-hour mark, with the scoreline 0-0, as increasingly heavy snowfall made playing conditions nearly impossible.
The referee called the players off the pitch as efforts were made to clear the snow.
However after a 25-minute delay, UEFA announced the match had been abandoned.
A message on the official Twitter page of UEFA said: "The #UCL fixture between @GalatasaraySK & @juventusfc has been abandoned after a snow storm caused the pitch to become unplayable."
A message on the mico-blogging site by Juventus said: "#GalaJuve has been abandoned due to snow. We'll bring you news on the rescheduled fixture as soon as we have it."
Under competition regulations the remaining match time must either be played on Wednesday, or on another date set by UEFA. The match clock stopped at 31 minutes.
Juventus need a point from their final game in the group to secure their ticket to the last 16 knockout phase along with leaders Real Madrid.
Olympiacos pip Benfica on head-to-head rule
Benfica were eliminated from this season's Champions League despite coming from behind to beat Paris Saint-Germain 2-1 in Lisbon in their final game in Group C.
Edinson Cavani gave the visitors the lead at the Estadio da Luz only for Brazilian forward Lima to level from the penalty spot before half-time.
Nicolas Gaitan then scored what proved to be the winning goal for the Portuguese giants in the 58th minute, but the final outcome was academic as Olympiacos got the victory they required in the night's other Group C game.
Manchester City fought back from two goals down to earn a 3-2 win at Bayern Munich and break the holders' record ten-match Champions League winning streak.
The German giants also suffered their first defeat in major competition since coach Pep Guardiola took charge in June.
Bayern also fell to their first home defeat -- after 18 wins -- since losing 2-0 at home to Arsenal in last season's knock-out stages of the Champions League in March.
With both teams already through to the knock-out stages, City avenged October's 3-1 defeat to Bayern at the Etihad Stadium, but still finished second to the Germans in Group D with both teams on 15 points.
Super-sub Tomas Wagner scored a last-minute winner to fire Viktoria Plzen into the UEFA Europa League on the back of a shock 2-1 Champions League win over nine-man CSKA Moscow.
It had been a campaign to forget for Plzen, who were pointless heading into Tuesday night's Group D game, but they knew that victory would be enough to see them pip CSKA to third place on their head-to-head record.
The Czechs were determined to score in the first half but failed to make the breakthrough - an honour which fell to Moscow's Ahmed Musa shortly after the break.
The game then swung on Alan Dzagoev's dismissal for violent conduct and Daniel Kolar promptly equalised to put Plzen back on course towards taking a first point.
All three would stay at the Doosan Arena, though, with Wagner rifling in a late strike before Pontus Wernbloom compounded CSKA's misery by getting himself sent off.