Despite their 200-mile journey south, it was the 2013 FA Vase champions who started the brighter side and took the lead after just 12 minutes through Nathan Fisher.
But Evesham came back into the game as the half wore on, with Adam Mann coming close just before the interval leaving it all to play for in the second half.
The FA Trophy
Second Round Qualifying
Saturday 14 November 2015
Winners receive £3250 from prize fund
Click here for results
By Jack Miller
And the Robins eventually got the equaliser they deserved ten minutes after the interval, with Carl Brown smashing a sensational 30-yard screamer into the net to send the home fans into raptures.
That sparked Spennymoor into life, and despite substitute Danny Frost putting them back in front, it was a nervy final ten minutes for the visitors after Shane Henry was booked twice in quick succession and got his marching orders.
The pressure mounted for an Evesham equaliser and Spennymoor racked up the bookings, but Ainsley’s side – seeking their 13th trophy in nine years – held on to progress and send two coaches of travelling fans on the long journey home with smiles on their faces.
Elsewhere, Kingstonian could not do the same and succumbed to a late goal at Dorchester Town.
Aaron Goode turned the ball into his own net early on to put the home side in front, and with Kingstonian struggling to get a foothold in the game the Magpies looked likely to double their advantage.
Great play from Ben Watson saw his shot deflected and tipped over the bar, before skipper Jake Smeeton also had an effort pushed away for a corner.
But against the run of play, the K's hit a quickfire double to go in ahead at the interval – Reece Beckles and Pelayo Pico Gomez scoring within a minute of each other just before the half-hour mark.
Dorchester threw everything they had at Kingstonian after the break, with Nathan Walker narrowly heading over from Bradley Tarbuck’s brilliant corner before the visitors shot themselves in the foot again with another own goal.
Dan Bennett was the unlucky man this time, levelling the scores and sending the tie into a replay next week.
There were 4-1 wins for Weymouth and Bognor Regis at Bedford Town and Taunton Town respectively, while Lewis Smith’s classy hat-trick handed East Thurrock United a 5-0 victory at home to Tiverton Town.
Buxton followed suit and racked up five goals at home to Radcliffe Borough, while Ciaran Kilheeney’s double helped Warrington beat Basford United 3-0, with the visitors reduced to nine men before half time.
Blyth Spartans also earned themselves a spot in the hat for the next round, beating Whitby Town 1-0 – Damen Mullen with a fine sole effort to dance through the defence before slotting the ball home.
And in arguably the tie of the round, Molesey scored three goals in the final 12 minutes to come from behind at home to Harlow Town and win 4-3.
The Hawks shot into an early lead when Jared Small headed home Ryan Melaugh’s driven corner from six yards out.
Melaugh was provider again on 28 minutes, whipping a dangerous free-kick into the box which Luke Elliott could only turn into his own net from close range to make the scoreline 2-0.
Molesey pulled one back two minutes later, as Stafforde Palmer produced a superb low finish into the corner but Leon Fisher restored Harlow’s advantage after half time when he tapped in from Layne Eadie’s cross.
The Moles looked down and out, but James McShane gave them hope 12 minutes from time and despite Niklas Freund’s heroics in the Harlow net, he could do nothing to stop Palmer stroking home his second of the afternoon seven minutes later.
The comeback was complete as the game entered added time, with McShane pulling back for Reis Stanislaus to leave Harlow’s Trophy dreams in tatters.