The Greenbacks proved superior to Sutton Coldfield Town for long spells in the preliminary round tie but in the end they were happy to hear the final whistle.
Goals from defenders Ian Roper
and Alex Gudger saw them coasting into a 2-0 lead at the end of a first-half they had dominated.
But a penalty provided a life-line for Sutton Coldfield before the sending-off of central defender Jamie Greenway
left the Greenbacks hanging on.
Bedworth twice went agonisingly close - hitting a post and the bar - before they snatched a 10th minute lead.
Top scorer Matt Gardner struck an overhead kick against the bar before the ball bounced out to stalwart ROPER who tucked away his first goal for the Greenbacks.
With goalkeeper James Martin ruled out by a shoulder injury, manager Steve Farmer
introduced his second stand-in goalkeeper in five days - Lee Windrum from Corby Town.
Windrum was called upon to make two excellent saves in quick succession from James Reid and Craig Marshall when both should have scored.
Sutton were made to pay for those misses as the Greenbacks notched a deserved second goal a minute from the break.
curled over a testing free-kick where on-loan GUDGER hooked the ball into the net when most onlookers thought it had gone out of play.
Bedworth, with ex-Nuneaton Griff midfielder Kyle Baxter
enjoying his first full game, continued on top and it seemed only a matter of time for a third goal.
But the Greenbacks were rocked by a 72nd minute penalty when winger Matt Dodd was brought down by Gudger and MARSHALL stepped up to cut the arrears from the spot.
Sutton broke clear again a few minutes later - and it led to them being reduced to 10 men as Jamie Greenway
was sent off for bringing down the fleet-footed Dodd just outside the box.
Tension then set in and it was the experience of veteran Roper - chosen as "man of the match" - that steered the Greenbacks through the closing moments with referee John Mayhew allowing six minutes of stoppage time.
Bedworth: Windrum, Dean, Gudger, Greenway, Roper, Baxter, Daniels, Ashley Spencer
, Gardner (Brown, 83mins), Piggon, Bates (Andrew Spencer
, 70 mins). Subs not used: Brehon, Moore.
Sutton Coldfield Town: Cooper, N Dodd, McNaught, Rathbone, Booth, Costello, M Dodd, Jones (Quiggin, 57mins), Marshall, Reid, Lennon. Subs not used: Milner, Miller.
Referee: John Mayhew (Dudley).