Last season's Bundesliga runners-up kept possession well throughout and scored three times in a topsy-turvy first half through Willi Orban, Emil Forsberg and Jean-Kevin Augustin.
Leipzig, who recovered from their defeat to Besiktas on the previous matchday, have four points, one more than their Portuguese opponents, who are third.
The big surprise in the Porto lineup was the absence of World Cup winner Iker Casillas, with the Spanish keeper being left on the bench in favour of Jose Sa, a 24-year-old making his debut in the competition.
The gamble backfired when Sa spilled a Bruma shot into the path of Leipzig captain Orban, who put the hosts ahead after eight minutes.
Porto levelled 10 minutes later when a deep throw-in was headed on to Porto danger man Vincent Aboubakar and he volleyed home for his third goal in two Champions League matches.
But Leipzig, with striker Timo Werner on the bench, struck twice in three minutes with Forsberg beating Sa, after a one-two with Marcel Sabitzer in the 38th, and 20-year-old France Under-21 international Augustin slotting in.
Porto reduced the deficit a minute before halftime but the hosts kept possession in the second half and saw out the match.
Augustin had two more golden chances to restore their two-goal lead, but their first-half goals proved enough for the Champions League newcomers to earn their first three points in this competition.
(Reporting by Karolos Grohmann; Editing by Toby Davis)
