Steven Gerrard's side kicked off their campaign in style last week with a 2-0 win over Standard Liege, but they looked destined for a disappointing draw at Ibrox until Morelos' introduction in the 63rd minute.
The Colombian made a swift impact after replacing Kemar Roofe, thumping home a close-range header five minutes later to seal an eighth consecutive win across all competitions for the Scottish Premiership leaders.
The result means Rangers are level on six points with Benfica – who they face next week – at the summit of the group.
Chances were at a premium during the opening 45 minutes of the first ever meeting between the two sides.
Roofe skewed an overhead kick wide in what was Rangers' best opening, while Poznan forward Mikael Ishak lashed over at the end of a largely insipid half.
The visitors started the second half brightly, with Thomas Rogne heading narrowly wide from Daniel Ramirez's free-kick.
Scott Arfield fizzed wide from 25 yards on the hour mark before Morelos met Borna Barisic's superb cross from the left with a powerful header from six yards that thundered past Filip Bednarek.
Poznan – who failed to register a shot on target – never looked like salvaging a point in the closing stages, with Gerrard's side seeing the game out in relative comfort for yet another win and a fifth clean sheet on the bounce.
What does it mean? Morelos decides dour affair
It looked like Rangers were set for the most uninspired of goalless draws before Gerrard turned to Morelos.
The 24-year-old did not let his manager down, powering home a decisive header soon after his arrival.
The hosts were not exactly a dominant force but, with Poznan failing to test goalkeeper Allan McGregor, they can certainly claim to be good value for all three points.
Davis' experience shines through
The midfield battle was not pretty but Steven Davis can reflect on a solid performance.
The Northern Ireland international made three tackles and gained possession nine times – both totals were more than any other Rangers player.
Roofe back to earth with a bump
The former Leeds United man lit up last week's win over Standard Liege with a stunning goal from 54.6 yards.
He struggled on Thursday, though, taking just one shot and completing a mere nine passes before he was replaced by Morelos.
Key Opta Facts
- Including qualifying rounds, Rangers have lost just one of their last 20 home games in European competitions (W14 D5).
- Rangers have won their last five matches in European competition, netting 14 goals and conceding just once.
- Lech Poznan have won just one of their last nine Europa League matches (D2 L6), winning 2-1 at Fiorentina in October 2015.
- Rangers have won each of their last four matches against Polish opposition in major European competition (excluding qualifiers).
Rangers travel to Kilmarnock on Sunday looking for a fifth consecutive Scottish Premiership win, while Lech Poznan visit Znicz Pruszkow in the Polish Cup a day later.