Alexis Sanchez gave Marseille the lead just after the half-hour mark from the penalty spot, before Sergio Ramos equalised on the stroke of half-time.
But Ukraine international Malinovskyi blasted the hosts back in front and they held on to end their long wait for a victory over PSG at Stade Velodrome.
"We know that our supporters have been waiting for that victory for a long time. We are happy to share this with them," Marseille captain Valentin Rongier told France 3 after only his side's second win in their last 27 meetings with PSG.
It was a second straight early Cup exit for record 14-time winners PSG after last season's last-16 loss to Nice.
"We made a few mistakes that cost us dearly," said PSG skipper Marquinhos.
"It's a defeat that hurts because it's a Cup match against our great rivals. We wanted to come back to Paris with victory and qualification.
"We have to do better, keep working and move forward."
Igor Tudor's Marseille will now be among the favourites to lift the trophy for an 11th time but first since 1989.
Marseille, who slipped eight points behind leaders PSG on Sunday after a 3-1 home loss to Nice, started slowly.
But they grew into the game after goalkeeper Pau Lopez had denied Nuno Mendes an early opener, with Matteo Guendouzi's shot excellently saved by Gianluigi Donnarumma.
Ramos was the culprit as Marseille forged ahead, clumsily fouling Cengiz Under just inside the area.
Sanchez stepped up to send Donnarumma the wrong way and score his 11th goal of a productive first season since joining OM from Inter Milan.
Lionel Messi, with a free-kick, and Neymar both came close to levelling, before Ramos did just that by steering a header into the far corner from a Neymar set-piece in first-half stoppage time.
Marseille restored their lead in the 57th minute as a loose ball broke to Malinovskyi on the edge of the box and the on-loan Atalanta midfielder hammered a left-footed strike into the top corner.
PSG rarely carried a threat but almost forced a penalty shootout when Neymar set Messi in the clear, only for the seven-time Ballon d'Or winner to fire off target.
Ramos had an injury-time effort chalked off for a clear offside and headed a good chance straight at Lopez as Marseille clung on.
Earlier on Wednesday, holders Nantes edged past Ligue 1 bottom club Angers on penalties after Florent Mollet scored an 87th-minute equaliser to force spot-kicks following a 1-1 draw.
Lyon also won a shootout against Lille to progress, having earlier blown a two-goal lead to draw 2-2.
Toulouse saw off in-form Reims 3-1, while second-tier clubs Rodez, Annecy and Grenoble all also booked their places in the last eight.