Joe Chalmers came within inches of an early opener for Falkirk after Craig Sibbald sent him clear, but his shot slipped just past the post.
Falkirk lost midfielder Mark Millar through injury after 16 minutes and were then fortunate not to fall behind.
Bob McHugh had a glaring opening in front of goal, but screwed his shot wide when it seemed he must score. Michael Paton then came close before half-time with a long-range effort which Michael McGovern diverted for a corner.
Phil Roberts carved out a great chance for Blair Alston early in the second half, but he dragged his shot wide.
A swirling wind caused problems for both teams and chances were by now few and far between.
Paton almost broke the deadlock with a stunning volley, but McGovern was equal to the task and kept it out.
Reilly finally put Queen of the South ahead with a header from Derek Lyle's cross after 81 minutes. Five minutes later, he scored his second which ended Falkirk's unbeaten 11-match run and knocked them off the top spot.