Histon's stunning victory over Leeds United on Sunday afternoon did not bring them the FA Cup third-round rewards they were hoping for.
The Blue Square Premier promotion chasers were 1-0 winners over League One giants United in the second round but were visibly disappointed to be handed a home tie with Swansea as they watched the live draw.
Fellow non-leaguers Blyth Spartans will host Premier League side Blackburn if they win their replay against Bournemouth.
FA Cup holders Portsmouth were drawn at home to Bristol City while Liverpool will travel to Preston, Manchester United face a trip to Southampton, Chelsea host Southend and Arsenal entertain Plymouth.
Histon's victory at a wet and wild Glassworld Stadium came courtesy of Matthew Langston, who headed in Gareth Gwillim's cross in the 39th minute.
Histon dominated the first half and could have gone into the break further ahead, but they had to withstand a barrage of pressure in the second half.
Lubomir Michalik twice came within a whisker of equalising for Leeds, first seeing his shot cannon back off a post then a fierce drive headed off the line by Jack Midson late on.
Manager Steve Fallon looked on the draw as a chance to get to the fourth round.
''It's a good draw for us,'' he told ITV1. ''Obviously, we wanted to be at home; that's all you want. You could have the big boys but Swansea are a big side in their own right.
''It's a good little tie for us and I'm sure we'll have a full house again and maybe a chance of getting to the fourth round.''
Talking about the performance today, Fallon added: ''It was a magnificent rearguard action in the second half. We had a bit of luck when we needed it but we were magnificent.
''We deserved that victory. This will do this club the world of good.''
Among the other stand-out ties in the third round, Coca-Cola Championship pace-setters Birmingham and Wolves will face each other at St Andrews.
The local rivals were involved in a pulsating 1-1 draw at Molineux yesterday.
Blue Square Premier team Barrow were rewarded for their 2-1 victory over Brentford on Friday with a trip to Middlesbrough while there could be an all non-league tie with Notts County or Kettering hosting Eastwood Town.
Northern Premier League side Eastwood caused the shock of the day yesterday with a 2-0 win over League Two leaders Wycombe while County and Kettering drew 1-1 on Sunday.
Brett Solkhon gave the non-league outfit the lead after 18 minutes but Sean Canham equalised 11 minutes before half-time.
Droylsden will travel to Ipswich if they win their tie against Chesterfield, which was abandoned yesterday because of fog with the Blue Square North side 1-0 up.
Three more Blue Square Premier sides are also in the third round with Kidderminster drawn away at Coventry, Torquay hosting Blackpool and Forest Green Rovers visiting Derby.
League Two Macclesfield were handed a glamour tie at home to Everton while last year's beaten finalists, Cardiff, host Reading.
Barnsley, who beat Liverpool and Chelsea last year, will have another chance to upset Premier League opposition when they travel to West Ham.
There are three all top-flight ties with Hull hosting Newcastle, Wigan travelling to Tottenham and Bolton visiting Sunderland, where they triumphed 4-1 in the league on Saturday.
Eastwood manager Paul Cox is a former Notts County player and was thrilled with the draw.
He told Sky Sports News: ''It's a dream tie for me, it's just fabulous, living in Nottingham. I'm keeping my fingers crossed now they actually beat Kettering. I think it's a winnable tie and it's a tie we'll be relishing.''