As a result of their loss, Milan have now won just once in their past seven matches across all competitions, meaning they are at risk of missing out on Champions League qualification.
Thursday's elimination at least allows them to fully focus on keeping hold of the fourth spot in Serie A, though Gattuso is well aware Milan are going through a difficult period.
"We held out well in the first half, but after half-time, they had three chances and we did not move enough off the ball," Gattuso told reporters.
"We're in a difficult moment. We're struggling and we need to turn the page because on Sunday there's another must-win game.
Gattuso on #MilanLazio : "We've suffered a heavy blow. Now, we must regain our energy and bring the fight to Torino"— AC Milan (@acmilan) April 24, 2019
Following a rough night, Coach Gattuso urges his boys to pick themselves up starting from Sunday's league clash in Turin: https://t.co/fe19TSDLjR #TIMCup pic.twitter.com/YU6GUNLllr
"We've got to try to understand why we're in this negative period. Lazio deserved to qualify [for the final]. Our fans deserved more.
"Sometimes you change just to do something different, to improve the performance, lift the team. Something isn't working and we've been struggling for the past month.
"I don't think that it is just a physical issue, but also mental and tactical, as we are performing below our potential.
"It's not an issue of systems or individuals, though the whole team has taken steps backwards. We are playing with the handbrake on and paying a price.
"We had tried to play a life-or-death game but it didn't work. There's no point denying it, we were embarrassed and have to look forward.
"Lazio were coming from a difficult period too, but we made it easier for them. We've got a month left in the season and it is our duty to give our all."
Next up for the Rossoneri is a trip to Torino on Sunday in Serie A, which is a must-win game for them to keep their place in the top four.
Gattuso's men currently occupy fourth on goal difference ahead of Atalanta, while AS Roma and Torino also still have their eyes on Champions League qualification, with five matches of the campaign remaining.