The Dee's 2-1 win in Paisley over St Mirren ensures their bid to beat the Clydesdale Bank Premier League drop will linger on until next week's home fixture with Hearts thanks to goals from Jim McAlister and Carl Finnigan.
Steven Thompson briefly levelled seven minutes into the second period when he thudded home a header at the back post. But after losing skipper Goodwin following two needless bookings, Saints found it harder to repel their desperate visitors, who finally struck eight minutes before the end with substitute Finnigan heading in off the underside of the crossbar.
Lennon admitted his team's chances had been hurt by the red card dished out by referee Craig Thompson.
He said: "Jim is an important and experienced player for us but he put himself and the team in a difficult position and Craig ran out of patience with him."
The Buddies boss was impressed by their opponents' fight and hopes his side can replicate it next week and end their own relegation fears before panic sets in.
"Dundee brought a fighting spirit and desire and that's something I'm hoping my players can learn from in the last four games," he added.
"When we bring our industry, it brings out our strengths. But when we don't, we are very ordinary. In fact, we were below ordinary.
"We can look over our shoulder now and see Dundee have picked up good momentum but we are still in a very comfortable position ourselves."