Hibs were made to work hard by their hosts but took charge after the break and have moved within two points of second-placed Rangers, whose match was postponed.
Queen of the South made a bright start and were unlucky not to take the lead in the fourth minute, when Iain Russell’s close-range effort came back off the post.
Ian McShane then shot just over the bar from the edge of the Hibs box after good work from Daniel Carmichael.
However, Hibernian then began to cause some problems for the hosts and a deflected Robertson shot looped towards goal and had to be headed off the line by Lewis Kidd.
Jason Cummings then found the net for the visitors but he had long since been flagged offside.
However, the hosts continued to look dangerous and Kevin Holt was just wide with a header from a McShane free-kick before Mark Millar had a shot saved by Mark Oxley.
Hibs then finished the half the stronger, Liam Fontaine having a header cleared off the line from Scott Allan’s corner and Martin Boyle shooting just wide of the target.
The second half began with vociferous calls for a penalty from the home players and fans after the ball appeared to hit Paul Hanlon on the hand but the referee was unmoved, and Oxley then saved another Millar effort.
Hibs then took control, however, and Cummings had a series of shots off-target before a Fontaine header was saved.
And Hibernian were then awarded a 63rd-minute penalty when Cummings was brought down by goalkeeper Jim Atkinson when he looked to be running too wide of the goal, and McGeouch found the bottom corner of the net with the spot-kick.
The vistors were keen to kill the game off and pressed forward looking for a second, Boyle hitting the post after a nice passing move involving Allan and McGeouch.
And they wrapped up the points when Robertson (75) scored a sublime second, receiving a neat lay-off and curling a lovely shot into the corner of the net from outside the penalty area.