Iain Russell then found himself one on one with Simonsen but smashed his shot straight at the Rangers keeper.
At the other end, Kenny Miller shuffled into the box and fired high and over the bar as Rangers looked to establish control in the game.
It was Queen of the South who went in front, however, when Holt curved a free kick into the net from outside the box.
An excellent chance then saw the hosts miss a great opportunity to double their lead as Gavin Reilly blazed over after a ball from the left had evaded everyone in the box.
Minutes later, Rangers almost grabbed an equaliser through Bilel Mohsni, but he headed just over the bar.