Celtic managed a 3-0 win over Motherwell at Hampden Park in the Cup final.
Lennon hopes the Cup triumph is the first of many as manager of the Bhoys.
“I want the big one on my CV and the sooner the better,” Lennon told Sky Sports.
“There’s always pressure at Celtic. Whether it’s bringing the championship back – we fell short of that – but this is the first trophy of many, we hope, for this squad.
“You can see the talent that we’ve got and they’re growing together and you can see the connection they’ve got with the supporters.
“I’m delighted for the players and the supporters, they’ve earned it and it’s a great way to end the season and gives us something to look forward to next year.
“It could have been five or six but we weren’t really clinical enough in the second half, but we were good going forward and scored some very good goals.”
He added: “We’ve had big games all season and played very well – seven Old Firm games – and we came out on top in the majority of them.
“We’ve gone other places and played very well, so I don’t buy into [the suggestion] that they’re not big-game players, because they are.”