Kaka should have added to their lead before keeper Iker Casillas also blocked shots from Thievy Bifouma and Sergio Garcia for Espanyol.
Midway through the second half Higuain hit a powerful drive into the top corner while Vladimir Weiss missed a good opportunity to reduce the deficit.
Substitute Jose Callejon slotted in a third against his former club and Higuain completed his hat-trick a minute from time as Real condemned Espanyol to their first defeat at home.
"I was happy with the way all of my players performed and also that of the opposition," said Real coach Jose Mourinho, who has complained in recent weeks that rivals have sought simply to defend against his team.
"The 4-0 scoreline was a result of our first two goals as they then had to attack and this extra space allowed us to score more. We are now only a point off the lead which is not a big problem."
Levante had Juanlu Gomez to thank for their 1-0 win over former leaders Real Betis, their fourth successive victory.
Promoted Betis won their first four games on the trot, but yesterday succumbed to their second straight defeat.
Arouna Kone had a couple of excellent chances to score either side of Juanlu's volley from a Valdo Lopes cross which broke the deadlock after 33 minutes.
Atletico Madrid drew 0-0 with Sevilla in a match between two sides who are expected to be fighting for a Champions League place by the end of the season.
The home side had the better openings but it will be a game remembered more for the number of fouls with the referee issuing eleven yellow cards.
Diego had a shot blocked by keeper Javier Varas and Jose Antonio Reyes missed the target with an excellent chance.
After the break Manu del Moral went close for Sevilla but Varas continued to save Sevilla while Radamel Falcao ballooned the best opportunity over the crossbar from inside the six yard box.
AFP