The in-form striker, who went into the game having scored five goals in his last six games, could have had a hat-trick had it not been for a combination of good goalkeeping and poor finishing.
The home side were nearly made to pay as Jem Karacan saw a fierce drive pushed onto the post by David Forde late on, but the game remained goalless.
Millwall made two changes from the side held to a 1-1 draw at Nottingham Forest on Saturday, with Tamika Mkandawire starting in the centre of midfield and Kevin Lisbie preferred to Neil Harris in attack.
Forward Simon Church came in for Brynjar Gunnarsson as Reading boss Brian McDermott made just one change to his line-up.
A torrential downpour just before kick-off continued during the early stages and made the playing conditions tricky.
Reading winger Jobi McAnuff had the first chance of the game but his long-range effort cleared the crossbar.
At the other end, Jimmy Abdou headed wide from six yards but it would not have counted as the midfielder was adjudged to be in an offside position.
Adam Federici was called into action for the first time in the 16th minute when Morison burst into the box down the left and got his shot away, but the Australian goalkeeper was equal to his effort.
Wales international Morison saw a cross-shot fly inches wide off the boot of team-mate Paul Robinson shortly afterwards as the Lions continued to make the early running.
Reading responded, though, and were denied the opening goal by the linesman’s flag in what was a very close call.
McAnuff played a raking cross-field pass to Jimmy Kebe on the opposite flank and he fired home superbly on the volley. Unfortunately the winger’s celebrations were cut short.
Federici came to the rescue again for the visitors after the Royals had been caught on the break, tipping a curling shot from Morison around the post after the striker had found himself one-on-one with the shot-stopper.
Shane Long, Church and Brian Howard were all involved in an almighty goalmouth scramble for Reading, while Morison was again denied by Federici on the stroke of half-time after being sent clear.
A quiet second half burst into life in the 62nd minute when McAnuff’s ball to the far post was met by Long but his close-range header was well blocked by Forde.
Kebe had a word with the referee after claiming he had been hit by a missile from the crowd as Reading pushed for an opening goal.
Howard forced a comfortable save from Forde with a 20-yard drive, before Karacan unleashed a piledriver from distance which was somehow tipped onto the post by Forde.
Kebe had the last opportunity of the game, but the keeper easily gathered.