Going into Tuesday's game on the back of their first defeat of the season, against Leeds on Saturday, Kenny Jackett's side barely got out of second gear as they brushed Boro aside.
Former Stevenage striker Morison, who impressed throughout, opened the scoring from the spot, before Trotter capitalised on some poor defending to double their advantage.
Scott McDonald pulled a late goal back through a David Forde clanger, but Millwall were worthy winners.
After a cagey opening period, the hosts had the first opportunity, but James Henry failed to trouble Forde.
That seemed to open the game up and, after both sides had half chances, the Lions took the lead courtesy of a generous penalty.
Matthew Kilgallon was the guilty man, bringing down Alan Dunne, allowing Morison to grab his second goal of the campaign.
Boro desperately tried to get back into the game with Tarmo Kink and David Wheater forcing saves from Forde.
But it was Millwall who got a second as Neil Harris headed Henry's corner back across goal, leaving Trotter with the easiest of finishes after the visitors failed to clear their lines.
Picking up where they left off, the home side made a lively start to the second period, with Morison and Henry impressing.
But, despite seemingly being unable to open up the Lions' backline, the visitors nearly fashioned an equaliser as Julio Arca's 25-yard strike struck the cross bar.
After Gordon Strachan brought on Kris Boyd in an attempt to galvanise his troops, McDonald grabbed a late consolation when Forde, thinking the ball was going over, bizarrely let the ball to go in.
But it did not matter as Millwall moved into the third-round draw.