Ajax, who are level on four points with AC Milan in second place behind Real Madrid on nine, made an excellent start when de Zeeuw scored after seven minutes with a fierce drive and Suarez doubled the lead four minutes before the break.
Auxerre got back into the match 10 minutes into the second half when Valter Birsa curled a free-kick into the net after Ajax's Andre Ooijer was sent off for a last-man foul on Dennis Oliech.
However, Oliech was himself dismissed five minutes from time when he handled the ball attempting to equalise and was shown a second yellow card.
In the final minute Auxerre substitute Jean-Pascal Mignot, warming up on the sideline, was shown a yellow card and then a red card in a matter of seconds by referee Olegario Benquerenca for continued dissent.
"Besides the phase after Ajax's red card we could not dominate at all against a better opponent," Auxerre coach Jean Fernandez said after his side's third successive defeat.
"We have lost all three matches but against bigger clubs than Auxerre. We still have home matches to play but it will be very difficult to finish higher than we are right now."
