Linfield started on the front foot and opened the scoring through Frizzell after just nine minutes.
Quick-thinking from Kyle McClean, from a free-kick, released the diminutive Scot who raced through a static Coleraine defence to blast his effort high past Lewis Webb in the home goal from 18 yards.
It could, and should, have been 2-0 moments later and, again, Frizzell was involved.
This time he whipped a cross in from the left towards unmarked striker Moore who headed it straight into the hands of the keeper from six yards.
The Blues continued to dominate and they doubled their lead midway through the opening half.
An Ethan McGee long throw was headed clear by a Coleraine defender only as far as Linfield captain Shields, who simply volleyed the ball into the roof of the net from 25 yards past a helpless Webb.
Coleraine, the league's highest scorers, failed to muster a shot on target in a frustrating 45 minutes for the hosts.
Head coach Ruaidhri Higgins range the changes at half-time and it seemed as if the move had worked when Shevlin headed home Ben Doherty's corner 10 minutes after the restart.
Referee Evan Boyce disallowed the goal, awarding a free-kick which goalkeeper Chris Johns took quickly.
The ball was then transferred to Moore who held his nerve to round his marker before planting a low shot into the corner of the net for his first goal in Linfield colours.
The hosts pulled a goal back with just under 15 minutes remaining through top scorer Matthew Shevlin.
The striker controlled a pass forward from Levi Ives before slipping the ball past Chris Johns from six yards for his fifth goal of the season.
The Bannsiders huffed and puffed in the closing stages but could find no way through a dogged Linfield defence as they suffered a first home defeat in 10 months.