De Zeeuw and Suarez both scored in the first half as Ajax dominated, but following the sending-off of defender Andre Ooijer after the break the visitors pulled a goal back through Valter Birsa but could not find an equaliser.
Two dismissals in the final five minutes compounded the visitors` misery - including one for substitute Jean-Pascal Mignot who never made it onto the pitch.
The Dutch side had the first chance on goal but when Suarez crossed from the right, De Zeeuw headed over the bar.
They went one better in the seventh minute when De Zeeuw unleashed a powerful rising shot from the edge of the area.
De Zeeuw and Mounir El Hamdaoui had been involved in the build-up but, after Olivier Sorin had punched out a shot from Rasmus Lindgren, the ball was not cleared and De Zeeuw shot across goal and into the top left corner.
Birsa tried his luck with a shot from long range but the ball was deflected wide for a corner as the visitors tried to get back into the game.
The corner was cleared but when it was headed back in Roy Contout was penalised for using his arm to fend off Ajax goalkeeper Maarten Stekelenburg before heading the ball into the net.
Five minutes before the break El Hamdaoui slid the ball into the channel and Suarez ran on and shot across goal with the outside of his right foot to double the lead.
El Hamdaoui could have extended the lead early in the second half but completely missed the ball when Suarez's free-kick found him unmarked.
Ajax were reduced to 10 men when Ooijer, who was the last man, pulled back Dennis Oliech on the edge of the area.
Birsa made no mistake with a left-foot free-kick which he curled over the Ajax wall to reduce the arrears 10 minutes into the second half.
Stekelenburg then did well to gather Oliech's shot but the Auxerre striker was pulled up for offside anyway.
Oliech was dismissed with five minutes to go when he was booked for a second time for punching a cross over Stekelenburg and into the Ajax goal to level it at 10 men apiece.
Auxerre substitute Mignot was then booked and dismissed as he complained to Portuguese referee Olegario Benquerenca about time-wasting while warming up on the sidelines.