That may seem odd to some after just one win in their last five Premier League games and Everton only one place above the relegation zone, albeit having games in hand on their rivals.
"We're very confident of staying up," Holgate tells Sky Sports ahead of Sunday's trip to West Ham. "I think the gaffer now and the team that he's brought in are unbelievable, I think everyone in the squad thinks that as well.
"I think it's a difficult circumstance that he's come into. But I think he's a great manager, and I think we are all going to benefit from him and as the season comes on everyone is going to see that.
"We've all got complete faith, he goes into every detail, and we know exactly what we need to be doing."
Holgate, in particular, has benefitted from Lampard's arrival. Both his place in the starting line-up and his self-belief have been restored under the former Derby and Chelsea manager.
Lampard has started Holgate in nine of his 10 games in charge. But it was a completely different story under Rafael Benitez, who in his 23 games on Merseyside handed Holgate the same number of starts.
Holgate, who describes himself as a "ball-playing centre half", has also been returned to his favoured position under Lampard - often on the right of a back three - after being used as a utility player in recent seasons.
"He's made quite a big impact on me," Holgate says of Lampard. "Just giving me the confidence of playing is something that's massive for me personally.
"To be a part of the team and playing every week, shows the confidence he has in me, so I am wanting to repay that."
Holgate is optimistic about the future at Everton under Lampard, but says "that's not what this season is about". He is under no illusions about the situation they find themselves in.
"We are in a relegation fight there is no denying that," he says. "We know where we are now, and we have 11 games to sort it out as a squad.
"If I am looking at other squads that are going to get relegated, I believe that we have the quality to get ourselves out of the trouble that we are in."
Of those 11 games, six come away from Goodison Park. And it's been a big problem for Everton getting results on the road this season - their last away win was at Brighton in August. No side in the league has accrued fewer points on the road.
Holgate admits that Everton's poor away record is "a little bit strange", but refuses to give the stats too much weight. For the defender, the important thing is "winning games, no matter where they are".
