The Barca and Argentina captain, 27, surpassed the record of 71 he jointly held with former Real and Schalke striker Raul when he put Barcelona 2-0 ahead in the 38th minute.
Luis Suarez had earlier opened his goal-scoring account in his sixth game for Barcelona after his £75million move from Liverpool.
Messi, who was playing in his 91st game in Europe’s top club competition, netted his second 13 minutes after the break before he put away his third with three minutes remaining to cap a 4-0 romp that kept Barcelona hot on the heels of Group F leaders Paris Saint-Germain.
Two goals from Uruguay striker Edinson Cavani and one from Swede Zlatan Ibrahimovic helped the French champions ease past Ajax 3-1 in Paris and maintained their one-point buffer over their Spanish rivals.