Ajax's Netherlands keeper Maarten Stekelenburg was at fault for the goal after his sloppy pass to Gregrory van der Wiel was intercepted by De Jong and the Twente forward calmly opened the scoring with a shot from a tight angle after eight minutes.
Champions Twente dominated the first half against the Dutch Cup winners and came close to doubling their lead when Theo Janssen hit the bar before Stekelenburg denied De Jong a second goal by keeping out his powerful header.
Ajax, who visit PAOK Salonika in the Champions League third qualifying round on Wednesday after a 1-1 draw at home earlier this week, were reduced to 10 men just before the break when Suarez was sent off for a flying tackle.
The Uruguayan was famously dismissed at the World Cup earlier this month for handball on the line in the last minute against Ghana. The Africans missed the penalty and lost in a shootout to send the South Americans to the semi-finals where they were beaten by Netherlands without the suspended Suarez.
In a lacklustre second half Twente, coached by Steve McClaren's successor Michel Preud'homme, gave Ajax no chance to recover but also wasted opportunities to settle the match early.
Twente open the Dutch season on Friday at Roda JC Kerkrade while Ajax travel to Groningen next Sunday.
