The hosts had two men sent off during Scotland's 4-1 UEFA Nations League victory.
Armenia showed signs of ill discipline after going 2-0 down at Hampden last week and looked like they could have folded if Steve Clarke's men had got another goal.
But they were well in the game at 1-1 when Arman Hovhannisyan committed two yellow-card offences inside 10 seconds with a late challenge on Nathan Patterson followed by a headbutt on McGinn.
Kamo Hovhannisyan was then sent off in stoppage time for a wild tackle on Lewis Ferguson and one Armenia player had to be dragged away at the final whistle as the home coaching staff raced on to calm their players down.
McGinn said: "It started to get a bit tasty but it was important that we looked after ourselves.
"A few of their lads lost their heads a little bit as the game was going against them.
"The boys were winding me up saying I was acting a wee bit too dramatic. I didn't expect him to put his head towards me but it was important that we looked after ourselves. He definitely caught me.
"As a team we had to be careful as the last game at Hampden we saw them getting a wee bit naughty in the last 10 minutes.
"So when you come to places like this you need to be prepared for it.
"Big Scott's (McTominay) booking aside, we dealt with all that very well."
Scotland fell behind inside six minutes and looked vulnerable in the opening half hour but ultimately won comfortably and the later draw between Ukraine and Republic of Ireland also helped keep them well in the hunt for top spot in Group B1.
"We could easily have crumbled," McGinn said. "The start we had was something the manager isn't happy with and rightly so.
"Even from the first minute when we kicked the ball out of play. That gives a team like Armenia some momentum.
"We had to show big character and we certainly managed to do that in the end.
"It was crucial that we ended on a high as it's been an extremely disappointing camp for us."