The Hoops boss had to endure a difficult night at a freezing Arbroath before the SPL leaders could add a fifth round William Hill Scottish Cup place to the last 16 Champions League spot they booked last midweek and the Scottish Communities League cup semi-final place which awaits them at the end of next month.
A terrific goal from Adam Matthews was all that separated the Hoops from the Irn-Bru Second Division side.
But Lennon believes this experience will stand all of his players in good stead for the next round tie, which is away to Raith Rovers.
He said: "I'm glad I went strong last night.
"We had to be physical, and we had to defend set-pieces well because it didn't look like a night we were going to cut them open.
"We flashed a few balls across the box, but we were wasteful in front of goal in the second half.
"But we have won the tie, and the objective was to be in all four competitions at the turn of the year, which we have done, so it's been a tremendous effort.
"It shows the determination of the players. They've been asked a lot of questions and risen to a lot of challenges. We are halfway through the season, and there is a lot more to look forward to.
"As it got colder, the pitch became a leveller. We were pretty much in control in the first half.
"But the pitch deteriorated very, very quickly, and the players found it very difficult to find their passing. The ball was just skidding away.
"So for the new guys to get a look at it, they will know now what British football at this time of year is all about.
"Even though Arbroath are two division below us, they gave us a hell of a game. Really, we should have put this tie away in the first game, and we are delighted we are through."