Colin Little's close-range strike in the 33rd minute was enough to give Alty a victory which secures them a handy £10,000 prize and the chance to repeat past Cup heroics when they square up to the 2004 beaten finalists at Moss Lane the weekend after next.
Heathcote said: "We weren't pretty, but we were effective and our reward is a fantastic draw against a well-known club who like ourselves, are having problems in the league.
"That said, the reputation of Altrincham FC is built on FA Cup success and this tie offers our current crop of players the chance to become part of this club's great Cup tradition."
Meanwhile, Droylsden chief Dave Pace is equally thrilled with the Bloods' prize for reaching round-one for the first time in 29 years and only the second time ever - a trip across the Pennines to meet UniBond division one north minnows Harrogate Railway.
Golden
A Terry Fearns tap-in and a Jody Banim penalty gave Droylsden a hard-fought 2-1 fourth qualifying-round triumph at Corby - their first away win since Easter Monday - to set up a tie which provides them with a golden opportunity to advance even further.
Pace said: "It was a relief to get through and, even though some people would have preferred us to have been drawn at home to a League club, I'm perfectly happy for us to be travelling to Harrogate Railway.
"It won't be easy, but, if we can go there and do a job like the one we did at Corby, we know we'll be one game away from being in the hat with the likes of Premiership giants United and Chelsea.
"That's an unbelievable prospect for a club of our size and, although we've a lot of hard work ahead if we're to achieve it, this draw offers us a fantastic incentive."
Crisis club Northwich Victoria, too, are through to the first round, in which they'll visit their Blue Square Premier rivals Oxford United, after sinking Southport 3-1 - their first win this season.
A Kevin Lee own-goal gave Vics a head-start before first Kevin Townson and then Paul Tait struck to provide them with some much-needed relief to their current on and off-field problems.
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