The Norwegians came into the game off the back of impressive victories against reigning champions Internazionale and Borussia Dortmund, and started particularly brightly.
The hosts were 2-0 up inside the first quarter of an hour, with both Morten Strand and Robin Bjornholm finding the net.
Liverpool never looked in control of the game and they were further behind on the half hour as Alexander Sorloth coolly slotted home to make it 3-0.
After the break, neither side were able to break each other down and there were few chances created.
The game finished 3-0, with Rosenborg deserved victors. Liverpool will host Rosenborg at Langtree Park in the reverse fixture which is now a must-win for the Reds if they are to qualify.