After a slow start to life in Germany, the legendary former Real Madrid and Spain striker added to his hat-trick against Werder Bremen with goals in the 30th, 50th and 87th minutes to take his season tally to 11 in all competitions.
Victory put Schalke 10th after five wins in their last seven.
Struggling Cologne stay 16th in the 18-team Bundesliga.
But Schalke pounced on a Cologne mistake soon afterwards when defender Christopher Schorch gave the ball away to Ivan Rakitic, whose cross was headed home by Raul.
Cologne went close just after the second-half restart but Clemens hit the side netting and, seconds later, Raul doubled the lead with a poacher's finish from close range.
But the veteran forward sealed the win and his treble with a late header from Edu's free-kick.