Cliftonville face a tough task in the second leg of the Setanta Cup quarter-final after losing to the holders.
Billy Dennehy fired Rovers into an 11th minute lead from Gary McCabe's cross before Barry Johnston went close for the Reds from 30 yards.
The hosts took a grip on the game in the second half and Colin Hawkins made it 2-0 from close range on 71 minutes.
Rovers keeper Oskar Jannson made a good save to deny Stephen Garrett as Stephen Kenny's side secured a deserved win.
Jannson saved from Martin Donnelly before Dennehy hit the opener at Tallaght Stadium.
Shamrock's keeper was in fine form and Janson kept out Eamonn Seydak's free-kick three minutes after the restart.
The League of Ireland champions began to threaten and Hawkins got the final touch to score from their first corner.
Rovers are in firm control ahead of the second leg on 20 March at Solitude.