Healy, ahead of his 500th game in charge of the Blues, called on his side to be better in possession in the second leg and they made a bright start in the south Belfast sunshine.
Two minutes in, Kirk Millar's deep free-kick was sent across the box by Ethan Magee, with Evan Caffrey alert to head the ball over the crossbar before Hall could get on the end of it.
Matthew Fitzpatrick then squandered a huge chance on 12 minutes.
Shels goalkeeper Conor Kearns' miscued clearance landed at the striker's feet, and he attempted to lift the ball over the goalkeeper, but his first-time lofted effort drifted wide.
Linfield were left to rue that miss when Coote opened the scoring against the run of play.
Harry Wood had a shot cleared off the line by Shields before the visitors recycled the ball to Coote, who stroked the ball into the bottom corner.
With their two-goal advantage in the tie, Joey O'Brien's side were happy to sit back and invite Linfield pressure.
They appeared to be coasting into half-time, until Linfield were awarded a penalty two minutes into first half injury time.
Tyreke Wilson handled the ball in the box and referee Andy Maddley pointed to the spot, with Shields keeping his cool to send substitute goalkeeper Lorcan Healy the wrong way.
McInroy thought he had restored Shelbourne's advantage by sweeping home after a well-worked short corner routine, but the goal was ruled out after a foul on Euan East and the Blues remained in the tie at the break.
