Hearts stayed in third after an impressive victory over Ross County and with the Jambos playing most of the second half with 10 men.
Callum Paterson got the hosts off to the perfect start with a stunning right foot drive from 25 yards.
The Jambos looked comfortable but then suffered a major blow when Blazej Augustyn was sent off for bringing down Liam Boyce when he was through on goal after an mistake by Igor Rossi.
However, Neilson has previously said that his side train with 10 men and it was evident on this performance as they swotted aside any pressure from the Staggies with relative ease.
“It was very disappointing. We’ve been getting a lot of plaudits recently and rightly so but we were well off it today and it wasn’t just one or two,” County boss Jim McIntryre said.
“We lost far too many individual battles first half and handed the initiative to Hearts. We were too predictable in the second half and it was just a frustrating day all round.
“We’ve got a chance to put this down as a bad day at the office and get ourselves into a semi-final.”
Hearts doubled their advantage on the counter with Jamie Walker playing an important role in the move which saw Osman Sow score his fifth goal of the season.
County had a late furry but never really looked like troubling Hearts who go into high spirits for the quarter-final League Cup tie against Celtic.
“It’s a great result for us. Ross County have picked up a lot of points recently and had a good result against Aberdeen,” Neilson said.
“I’ve said before and this got me into trouble that we do practice with 10 men. It’s about keeping their discipline and they did that today. They kept their shape, were compact and showed them wide. It was a good performance.
“We’re far far away from Celtic at the moment. We’re not getting carried away after back-to-back wins.”