Cavan had led for most of the match but McManus was played through to score the game's only goal on 59 minutes.
However, Cavan almost snatched a replay with the last kick of the match but Ryan Connolly's shot smacked off an upright in stoppage time.
Monaghan will play Down in the semi-finals on Saturday, 24 June.
Manager Malachy O'Rourke has led the Farney County to the Ulster semi-finals in each of his five years in charge, winning the Anglo-Celt in 2013 and 2015.
Cavan will now play in round 1B of the All-Ireland qualifiers on Sunday, 25 June.
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With a strong breeze at their backs, Cavan led 0-9 to 0-7 in the quarter-final at Kingspan Breffni.
Niall McDermott and Cian Mackey landed two points each while McManus scored four, two from frees.
There was just one point in it, approaching the hour mark, when home keeper Raymond Galligan made a superb save to keep out Jack McCarron's goal-bound effort.
From the 45, Monaghan keeper Rory Beggan stroked over a point to level at 0-13 each.
Then came the goal with class act McManus making no mistake after getting through to put Monaghan 1-13 to 0-14 up.
A Seanie Johnston free in stoppage time put Cavan three behind and they came close to levelling with the final kick when young Connolly's shot came off the post.
A second-half strike by Conor McManus proved the difference as Monaghan beat Cavan 1-15 to 0-15 in the Ulster Championship quarter-finals.