The 22-year old Danish striker scored the goal in injury time when he controlled a fine pass from Wout Weghorst, broke free on the edge of offside and calmly slotted the ball home.
Before that Weghorst had turned the tide in Ajax's favour when he gave them the lead after 68 minutes as he scored on the rebound after Fortuna goalkeeper Luuk Koopmans had saved a penalty from Steven Berghuis.
Ajax had otherwise struggled in the first half on a difficult pitch where they were fortunate to see an effort from the home denied by the woodwork.
In the second half, Fortuna Sittard struck the post again when Syb van Ottele fired a long-range effort at Ajax’s goal, but his shot crashed against the crossbar.
It was game number 21 for Rasmussen who joined Ajax from FC Nordsjælland in 2019.
Ajax and PSV are level on points, but the Amsterdam side have played one game less.