Martens scored twice in the first half to give Barca a two-goal aggregate lead following a 1-1 draw in the first leg in Paris. Marie-Antoinette Katoto got a goal back for PSG before the break but the French side could not find the second away goal that would have taken them through.
Lluis Cortes' side have been dominant domestically, winning all 25 of their league games so far this season, but are desperate to take the next step in Europe. They lost to Wolfsburg in the semifinal last season and were well beaten by Lyon in the 2019 final.
There will be no Lyon in Gothenburg on May 16, though. The seven-time winners of the competition -- including in each of the last five seasons -- were knocked out by PSG in the last eight. Barca will instead face either Chelsea or Bayern Munich.
PSG, who like Barca have never won the Champions League, will now focus all their energy on knocking Lyon off their perch in France. They are yet to win Division 1 Feminine but can move ahead of Lyon at the top if they win their game in hand. The two teams still have to play each other in the run-in, too.
A crowd of 1,000 was permitted at the Johan Cruyff stadium to see Barca make a fast start. Martens opened the scoring in the eighth minute with a brilliant finish after being played in by Leila Ouahabi.
The Dutch forward added her second just after the half-hour mark when she finished from close range after good work from Caroline Graham Hansen.
Barca's two-goal cushion didn't last long. They failed to deal with a PSG corner and Katoto was quickest to react to the loose ball in the box, stabbing home from six yards.
Prior to Martens second goal, the French side had been pressing well without really creating anything and that pattern continued into the second half. Barca saw a penalty appeal turned down after Christiane Endler spilled the ball and got tangled up with Alexia Putella, while Sandra Panos saved smartly from Ramona Bachmann at the other end.
Hansen then had a shot tipped over following a brilliant solo run and from the resulting corner, Jenni Hermoso hit the post. Marta Torrejon then headed against the bar in the series of corners that followed.
Led by the impressive Maria Leon, Barca continued to deal with everything PSG had to throw at them and should have killed the tie off earlier. Substitutes Mariona Caldentey and Asisat Oshoala both missed chances to put the result beyond doubt.