The Blucerchiati still have a two-point lead over Palermo in fourth place and go head-to-head in Sicily next Sunday.
"Today Livorno fought hard to honour their season, even if they were already relegated," Del Neri said of a 2-0 win.
"Our only course of action was a victory and it is not easy to play with that kind of pressure on our shoulders.
"Of course our attitude and concentration will have to be even more intense in Palermo, but that is a match I don't have to motivate anyone for. It prepares itself.
"These two teams have both had an extraordinary campaign and gone far beyond expectations.
"Both would deserve a shot at the Champions League and it's great to be battling it out just two weeks from the end of the season."