Former Vase opponents Vauxhall Motors provided the Hornets with a stern challenge for the second consecutive season and came within two minutes of pulling off a shock win at the Green Energy Sports Ground.
All was going to plan for Bolam and his players as Michael McKeown fired in them in front on 15 minutes and that lead remained intact until the last quarter of an hour.
But the visitors rallied and turned the tie on its head as an equaliser from Hayden Cooper and a late Michael Burkey goal gave them an unexpected lead in injury-time.
However, there was to be more drama as Hornets full-back Dan Groves saw his cross-shot find the net to send the tie to penalties.
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A Mark Foden save gave Graeme Armstrong a chance to seal a 5-3 win in the shoot-out and the veteran striker made no mistake to secure his place in Monday’s fourth round draw.
After the game, Bolam told The Gazette: “As much as I was disappointed with the performance, I can’t fault the resilience and the commitment of our players.
“That attitude runs right through the group and that showed in midweek in the Durham Challenge Cup win against Gateshead Rutherford too.
“We are finding a way to come through games and that was evident on Saturday more than ever.
“The lads know they haven’t played well; they have not found the intensity or energy that we expect of them.
“We did it the hard way, but I don’t want anything thinking I am down about anything today because the resilience was excellent.”
The win keeps alive Hebburn’s hopes of reaching consecutive FA Vase Finals as the prepare to head into their Covid-enforced Christmas break.
News is still awaited over when last season’s final against Northern League rivals Consett will be played with a date in late-March 2021 believed to have been discussed by the Football Association.
Saturday’s win over Vauxhall means that Bolam’s men are now just four wins from a second appearance at Wembley – but the Hornets boss insisted that his players will not be carried away by the prospect.
He said: “We have a break now and the players will have Christmas and New Year off, but we will try and have a couple of training sessions because we know we have to hit the ground running when we return.
“Our next game looks like it will be the next round of the Vase, which is unusual and that takes the cup feeling away – but that is how this season has been, it’s unusual but we just have to get on with it.
"The Vase is a competition that we have done very well in, but we aren’t getting too carried away.
“Wembley is a very long way away and we still have plenty of work to do because we start thinking of that.”