Guinea international Guirassy, who has scored 16 times in the league this season, combined well with Enzo Millot to slot the ball in through the legs of keeper Gregor Kobel in the 54th minute for his second German Cup goal of the season.
Stuttgart should have taken the lead much earlier and wasted a string of chances, with a toothless Dortmund reduced to defending and rarely venturing forward.
The hosts did score a deserved second goal in the 78th minute with an almost identical move to their first as Silas charged into the box to beat Kobel.
It was the second time in less than a month that Stuttgart, third in the league, beat Dortmund after their 2-1 win in the Bundesliga in November.
Dortmund's disappointing evening was completed when defender Julian Ryerson was helped off the pitch in agony after a fall in stoppage time.
Earlier Bundesliga leaders Bayer Leverkusen secured their spot in the next round with a 3-1 victory over second-tier Paderborn.
Eintracht Frankfurt, however, crashed out of the competition, losing 2-0 to third-tier Saarbruecken, who had eliminated Bayern Munich in the previous round.
