The striker struck the winner on 66 minutes when a ball from Joe Colbeck was played over the top of a static Chester defence.
They appealed in vain for the offside flag but Conlon flicked past the exposed John Danby.
The strong wind made good football difficult but it was the Bantams who seemed to cope better.
They had a great chance to score on 24 minutes when Tony Dinning's back pass was picked up by Danby.
The resultant free-kick on the edge of the six-yard box was hit powerfully by Paul Evans, but the Chester keeper made amends with a fine save.
A swirling ball from the right flank had Danby in trouble just before the break.
The keeper at first came off his line in an attempt to claim the cross but the wind held it up presenting it to Paul Evans.
But with the keeper in no man's land the midfielder headed well off target.
With the wind at their backs Bradford put pressure on the Chester goal early in the second period.
Conlon saw his goalbound shot deflected over for a corner and, on 56 minutes, Kyle Nix beat the Chester keeper to the ball and headed an effort goalwards with defender James Vaughan arriving in the nick of time to clear off the line.
Following Conlon's goal, Kevin Ellison should have levelled matters on 74 minutes with a free header from only three yards, but he directed his header straight at keeper Scott Loach when he looked odds on to score.
Conlon almost added his second when his spectacular overhead kick cannoned back off the bar ten minutes before the end.