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The Lowland League side take on East of Scotland League outfit Lothian Thistle Hutchison Vale for the right to host Celtic in the competition's last 16.
"I was sitting with my wee girl about to change her nappy. I just jumped up and went mad," said Russell.
Asked where the nappy ended up, he replied, "I don't know!"
The centre-back said he told his girlfriend, who was also watching the draw being made at Hampden, he was certain that when his own team and Vale's names were pulled out that Celtic's would follow.
However, the 26-year-old plumbing and heating engineer confessed he might struggle to focus on his job in the coming days.
"I think my boss will understand if I try to put my head down a toilet or something," he joked.
"I tried to go the gym. I was sitting on the bike just shaking my head."
Russell and his team-mates are determined to focus on their rearranged home game at K Park against Vale on 20 January, but the prospect of the Scottish champions playing at their tiny Lanarkshire ground inevitably seeps into the conversation.
So, can he imagine Celtic manager Ronny Deila watching him play from the stand?
"I'll put it out my head and make sure we get the win," he replied matter-of-factly.
"But after the game if he wants to talk contracts I'll be free!"
Sean Winter was also at home watching Monday's draw.
The former Stranraer midfielder told BBC Scotland: "It was me, my wee boy and my girlfriend and I just had a feeling. I've had a feeling for a few weeks it was going to be Celtic.
"I've played Rangers but I've never had a chance to play Celtic. As soon as I saw East Kilbride, I knew the next one was going to be Celtic. I was absolutely delighted.
"Whether it's us or Lothian Thistle, we have earned the right to be there. It's a fantastic thing for the town."
Fellow midfielder Craig Hastings, who works at the town's tax office, said: "I'm excited. I was sitting there with my wee maw, watching the telly, and she said, 'Who do you think it will be?'
"When Celtic came out she started screaming and then the phone was red hot, with my dad and my gran and everybody phoning me.
"Everyone is talking about it but we're not there yet. It's a great draw if we get there. We've got to prepare well."
Matt McGinley, a PE teacher at Calderglen High in the town, added: "It's the draw we wanted, the big-money tie, we are all absolutely buzzing."
"Bring them on. You want to test yourself against them the best, so why not?"
East Kilbride captain Barry Russell admits that seeing Celtic being drawn on TV as potential Scottish Cup opponents led to a domestic mishap.