It was a busy start to the game with Odhran Casey clearing Offord's effort off the line and Jack Keaney nodding Rory Hale's corner wide at the other end.
Linfield broke the deadlock just before the 30-minute-mark when Cliftonville failed to take the ball off Offord, despite multiple attempts, with the striker sending a strong strike into the top corner from outside the box.
A fingertip save from Morrison then denied the Blues a more comfortable lead seconds after the opening goal with the Cliftonville stopper diverting McClean's free-kick onto the left-hand post.
David Healy's side continued to dominate and had a good chance on the counter-attack but after a good move from McClean in midfield, Offord fired wide.
It was the away side who had the final chance of the half with Chris Johns called into action to thwart Thomas Lavery's low shot in additional time.
Just after the restart, a mix-up between Johns and Sam Roscoe nearly gifted Eric McWoods a chance, but the Blues were able to clear their lines before the Cliftonville forward could connect with the ball.
McClean then curled an effort just wide of the left-hand post before Hale's shot was comfortably gathered by Johns.
In the closing stages of the match, Matthew Fitzpatrick and substitute Callumn Morrison fired wide for the Blues, who now go into the second leg of their Uefa Conference League play-off against Shelbourne on Thursday looking to overturn a 3-1 deficit.