Saints travel to Dumfries on Tuesday night looking for their first win of the season against a team who suffered their first defeat on Saturday when going down to Hamilton in the Scottish Championship.
The third-tier champions and Ramsdens Cup holders sprang several surprise results last term and Lennon knows St Mirren will have to be at their best to avert a second-round shock at Palmerston.
Lennon said: "Winning the cup last season was such a highlight for everyone associated with this club. It was only the fourth time in 134 years that this club has won a national trophy.
"The celebrations were great but nobody hands out trophies for free. We know how hard we worked last year to lift that trophy and we need to work hard to have a chance of doing it again.
"It's a massive one for us. Everyone wants to knock out the holders of a trophy. Look at last season when the holders, Kilmarnock, were put out by Stenhousemuir. We certainly don't want to suffer a similar fate.
"We will not be able to drop our standards for even a second against Queen of the South. We need to put in a top performance."
Saints have shipped three goals in each of their two Scottish Premiership away games this season - against Ross County and Inverness - and even more when they played on the new artificial surface at Palmerston before the campaign started.