Gudmundsson struck the post from 25 yards before latching onto Morgan Fox's cross to put his side ahead.
Ademola Lookman fired past Marco Silvestri to double the Addicks' lead after the break.
Leeds' Chris Wood then struck the post with a header before Sol Bamba nodded home to reduce the arrears, but League One-bound Charlton held on.
The home side went into the game on a better run of form, having gone four league games without defeat, but the mid-table side lacked intensity from the outset.
Their first real chance came in the second half courtesy of Wood's header which came off the woodwork, before Bamba's free header briefly gave them hope.
But Charlton, whose relegation from the Championship was confirmed on 19 April following a goalless draw with Bolton, looked bright throughout and had Nick Pope to thank for ensuring all three points after he denied Mirco Antenucci with a fine injury-time save.
Steve Evans on his future as Leeds head coach: "You never hope one result is the difference between whether you are kept or not.
"I had a very good meeting with the president (Massimo Cellino) yesterday and it went very well. There were a lot of positive, constructive things said about myself.
"I gave my views on what we need to win promotion next season and he gave his."
Charlton manager Jose Riga: "In the first half we were in control. Leeds reacted for 15 minutes or so, which is normal, but we built a two-goal lead.
"After Leeds scored we defended well, especially our goalkeeper Nick Pope. I told the players that so many things have happened at our club but what matters is what happens on the pitch.
"I told them they had to show their quality collectively and individually, which they did."
Johann Berg Gudmundsson scored for the second successive game as already-relegated Charlton overcame Leeds.