Fitzgerald was on target again for the Cards in a dominant third quarter, with his touchdown following a two-yard interception from Dan Williams to put the visitors 24-13 ahead.
Back came the Rams early in the fourth quarter, with Cook finishing an 80-yard drive that Sam Bradford turned into a two-point conversion to give the hosts hope.
Zuerlein then made a 38-yard field goal to tie the game at 24-24 with nine minutes left in the fourth quarter, as both sides were struggling to force a late breakthrough.
The Cards failed to make a pair of late possessions count soon after and were punished by Zuerelein, who was given a late chance to snatch victory by a strong Rams drive.
With three field goals to his name Greg 'The Leg' lined up a strike from 48 yards with 40 seconds on the clock and made no mistake to help St Louis win their opening game for the first time in six years.