Luis Suarez put Ajax ahead just before half-time but a second-half brace from Blaise N'Kufo - the first a penalty - turned the tide and give Twente a surprise advantage heading to the Amsterdam ArenA next Sunday.
After kick-off was delayed 10 minutes due to crowd congestion at the Unive Stadion - Twente's temporary home - the game got off to a slow start and neither goal was truly threatened until late in the first half.
After Romano Denneboom hit a weak shot on goal for Twente, Ajax responded when Klaas-Jan Huntelaar met Rasmus Lindgren's cross but sent his effort just wide.
Ajax took the lead with a minute remaining in the half as Twente squandered possession and Kenneth Perez played the ball into the path of Suarez who finished from the edge of the area.
Twente looked a different team in the second half, and they attacked with far greater purpose.
Denneboom forced a fine save out of Maarten Stekelenburg in the 51st minute with a shot that looked to be heading for the top left-hand corner of the goal.
Twente drew level in the 64th minute after Ajax defender Jan Vertonghen handled the ball inside the area. N'Kufo stepped up to hit an unstoppable shot into the top left-hand corner of the goal.
And the hosts completed the turn around when Youssouf Hersi and Stein Huysegems combined to set up N'Kofu who again fired into the top left-hand corner.
The goal capped a fine second-half performance that will give Twente belief that they can complete the job next weekend.