Shearer's double took his tally to 18 for the season and 191 for Newcastle - just nine adrift of Milburn's record.
United cruised to a 4-0 win and a spectacular 7-1 aggregate victory against the Greek League leaders.
So now United go into Friday's quarter-final draw in Nyon on the back of eight successive wins for the first time in 11 years – the start of the 1994-95 Kevin Keegan promotion season.
Following a lacklustre first quarter-of-an-hour, the woodwork came to the rescue of the Greeks to prevent one of those most rare of commodities - a Nicky Butt goal - when his 25-yarder rattled to safety off the bar.
But that was only a temporary setback as United stormed ahead three minutes later with the most precocious of goals from Dyer.
From their first foul of the evening Olympiakos were punished. Laurent Robert's piledriver was blocked and headed back into the danger area by Lee Bowyer. Dyer – with his back to goal – then nonchalantly back-heeled the ball into the net, beyond the stunned Antonios Nikopolidis.
The game's first corner almost produced a second goal on 27 minutes when Steven Taylor meet Robert's cross but the youngster could only direct his header into the side-netting.
The alarm bells sounded in the Olympiakos rearguard on the half-hour mark when Shearer got clear of Georgios Anatolakis and from just inside the box fired high over the bar.
Shearer was then just inches away from turning a Robert free-kick home as United piled on the pressure and he went even closer with a glancing header that flew agonisingly wide from another Robert cross.
As they say, you can't keep a good man down and Shearer proved that point on the stroke of half time as Dyer turned from scorer to provider.
A clever break on the right saw the ball pulled back for United's record European marksman – that's 27 for the legendary striker – whose shot took a deflection into the net.
Nine minutes into the second half, United repeated the three-goal haul that they achieved in Athens last week and a 6-1 aggregate lead.
Nikopolidis pulled off a stunning save to prevent a fluke Bowyer header crossing the line but the ball fell to Jermaine Jenas who teed up Bowyer and his shot at the second time of asking squirmed under the keeper's body and over the line.
A totally deflated Olympiakos finally managed their first shot on the hour mark which failed to trouble Shay Given before United went four up after 69 minutes.
Jenas gobbled up the ball in midfield and slipped in Shearer who was played onside by Dimitrios Mavrogenidis and went on to round the keeper and fire into the net.
It could have been a lot more as United sprayed the ball about content with their night's work.