The 36-year-old died on the morning of Saturday’s semi-final with Hibs, having battled motor neurone disease (MND) for the past few years.
Neilson revealed that he found out about the Lithuanian’s passing just moments before kick off, with Hearts going on to win the match 2-1 after extra time to set up a December date with either Celtic or Aberdeen at Hampden.
Zaliukas played for Hearts between 2006 and 2013 and was the skipper who lifted the Scottish Cup when the Jambos defeated Hibs 5-1 in the 2012 Scottish Cup. The Kaunas-born defender, who won 25 caps for his country, also played for Rangers, Leeds, FBK Kaunas and Zalgiris in a distinguished career. He played more than 200 games for Hearts and scored 11 goals, including a winner in a 2007 derby at Easter Road.
Neilson was a team-mate of Zaliukas for three years when they were together at Tynecastle and the current Hearts boss was quick to pay tribute to Zaliukas in the aftermath of the win over Hibs.
“Firstly, it’s absolutely devastating for everyone at the club,” said Neilson. “I found out a minute or two before the start of the game.
“I didn’t speak to the players about it, just after the game.
“I knew Marius very well, he was here as a player with me and I knew him for a number of years.
“It’s devastating for everyone at the club because he is synonymous with the 5-1.
“Hopefully it’s fitting ... another game against Hibs and a victory at Hampden. He is in all of our thoughts tonight.
“Yes, we are celebrating and happy about getting the result, but we are also feeling it for Marius’s family because he was a top guy and I had a great relationship with him.
“He has had a difficult couple of years.”
When asked if Hearts would now try to go on and win the trophy for Zaliukas, Neilson added: “One hundred per cent. I would take another 5-1 win, whoever we play against, that would be great.”