Lennon's future was still a topic of speculation when Celtic went into last year's final against Motherwell after losing out to Rangers for the other two domestic trophies.
But a 3-0 win over Motherwell sealed his first silverware as manager and gave many of his players their first winner's medals too.
Celtic travel to Peterhead on the back of 10 consecutive league wins and Lennon is keen to ensure they maintain their momentum in the fourth round tie at Balmoor.
"It's just part of the domestic scene, sometimes it's nice to have the change from the rigours of the championship," Lennon said.
"We're the holders of this competition and we dearly want to win it again.
"The lads got a taste of winning something, for the majority for the first time, so that gives them a big incentive to win it again."
Lennon knows only too well the dangers of this cup competition, having been on the end of two of Celtic's biggest shock defeats, against Clyde as a player and Ross County as manager.
"There should be a gulf in quality but anything can happen on any given day," he said.
"We have to be ready for that. It's not as if we can just turn up and play 90 minutes and go home. We have to earn the right to play and win the game.
"This will be a huge game in Peterhead's history. These players will be motivated to take our scalp and we have to be very careful."
Celtic go into the game without Gary Hooper, who will be rested after playing through a groin injury.
Anthony Stokes will be Celtic's main goal threat in his absence but Stokes believes the crucial element to Celtic's turnaround in fortunes over the past two months has been tightening up defensively.
"The start of the season especially we were leaking goals and we weren't playing particularly well," the former Hibernian striker said.
"But even in the last five or six weeks there have been one or two games we have had our patches where we have gone through bad spells in the game, but we haven't conceded like we were at the start of the season.
"The defence has been solid and it's been good for us up front, because we know if we score in games there's a good chance we'll win."