Sisto struck the only goal of the game in the 69th minute, four minutes after Panama forward Blas Perez was shown a red card for a dangerous high challenge on Kasper Schmeichel as the Danish keeper left his area to head the ball clear.
The hosts took advantage of the extra man with Sisto given the freedom to go on a mazy run through the Panama defenders stretched across the edge of the area before firing home.
The Danes, having played with a more direct approach in their first qualifying campaign under Norwegian manager Age Hareide, turned on the style against Panama, who will be at their first World Cup when the Russia finals start in June.
Yussuf Poulsen wasted a golden chance to put the hosts in front early on, shooting straight at keeper Jaime Penedo in a one-on-one created by a superb pass from Nicolai Jorgensen.
Sisto's dribbling skills and variety of movement regularly brought the home fans to their feet in the first half, but the Danes struggled to break down the determined visitors.
Michael Murillo had Panama's best chance of the opening period, but he could not get enough power behind his hooked volley to trouble Schmeichel.
There were plenty of strong tackles during the match but Perez was dismissed for his high challenge on the keeper.
Sisto, who plays for Spanish club Celta Vigo, finally gave the hosts the lead, cutting into the box on his right foot before curling the ball into the far corner of the ent.
Denmark meet Peru in their opening World Cup Group C game on June 16 before facing Australia and France.
Panama have been drawn in Group G and open their campaign against Belgium on June 18 before meeting England and Tunisia.
(Reporting by Philip O'Connor in Stockholm; Editing by Ken Ferris)
