Austria's Konrad Laimer was sent off in the 36th minute, but Tunisia could not take advantage of the sustained numerical advantage.
Sweden's opening opponent on June 15 in Monterrey had three shots on target in the first half. Instead, the home team got the upper hand. In the 63rd minute, Marcel Sabitzer scored to widen the lead with a shot near the right post from a pass by Stefan Posch.
Austria is playing in a World Cup group with Jordan, Algeria and Argentina.
Tunisia will face Belgium on Saturday in Brussels before departing for the World Cup.
