The Easter Road men never reached the heights in this attritional encounter but were still able to grind out the win that maintains their advantage at the top of the Championship table.
With Lennon calling the shots from the dugout, Shinnie, on loan from Birmingham for the season, is convinced there is more to come.
He said: “He has managed at the highest level so he is demanding as a manager but I like that. It gets the best out of us. The manager now won’t let us off with slipping up at places like this – he won’t accept that.
“So everyone is on their toes and if you don’t perform then you’re off and you could be out the team and I think it’s showing with our performances right now.”
Shinnie was expecting a tough encounter and said results like this show the difference in the Hibs side this year. He added: “It was a tough game – it was always going to be. The gaffer had drummed that into us all week. A few teams have come undone here this season and we were last year, so it was always going to be tough but we were up for the battle and I think that’s the difference in us this year.”
While Hibs maintained their perfect start in the Championship, they also kept a third consecutive clean sheet and Shinnie said: “The boys at the back were brilliant. They had to deal with a lot of long balls and they did it really well.
“Unfortunately we just couldn’t get that second goal that would have killed the game off but we said at half-time if we have to win 1-0 then we will and that’s what we did.
“If we keep a clean sheet then we should score in every game with the attacking players that we’ve got.
“It was a penalty today, and Jason [Cummings] maybe could have had a couple which would have made it comfortable, but we won and that’s all that matters.”
Hibs scored the only goal after 32 minutes when Jason Cummings converted from the spot to cap his 100th start for the club after Grant Holt was fouled in the area.
Despite spells of dominance, Hibs were unable to increase their lead against a dogged Dumbarton side, whose manager, Stevie Aitken, said: “I thought the penalty was harsh and it’s difficult to take because that’s the difference between us and Hibs today.
“But we can take a lot of heart because it was a great performance from my guys.
“Our season won’t be defined by these games. It’ll be on the games against teams around us.”