Both sides had made good starts in the group by winning their opening games two weeks ago, Porto beating Dinamo Zagreb 2-0 in Croatia and Paris running out 4-1 winners at home to Dynamo Zagreb.
But it was Porto, whose four-match winning run was brought to an end in a domestic draw at Rio Ave last Saturday, who started better with Rodriguez testing visiting goalkeeper Salvatore Sirigu in the third minute.
The visitors, heading into the game on the back of five straight wins, looked nervous in the early stages and Porto had another chance after five minutes when Joao Moutinho's right-foot shot flashed just wide.
Porto had more chances in an attacking start, Rodriguez failing to test Sirigu from long-range and Brazilian Danilo bringing a save out of the busy keeper all before 10 minutes of the match were gone.
Zlatan Ibrahimovic headed the visitors' first chance just wide after 12 minutes before it was Porto who threatened again, Moutinho shooting wide then Lucho Gonzalez seeing a long-range effort blocked in the box.
For all Porto's dominance, it was Ibrahimovic who had the best chance to break the deadlock in the 22nd minute when the Swede's rising shot from the edge of the 18-yard box was tipped over by Porto keeper Helton.
Six minutes later, Sirigu produced a fine save at the other end to tip a Rodriguez chance round the post, then Jackson Martinez met a Moutinho free-kick into the box with a header which flashed inches wide.
Paris finished the first half with a good chance for Mamadou Sakho, who headed just over, but the hosts continued to look the more threatening at the start of the second period with good chances for Martinez and Rodriguez.
The Ligue 1 side responded with both Nene and Thiago Silva shooting over the bar, but the action swung back in Porto's favour with Sirigu saving with his feet to deny Silvestre Varela before Moutinho fired the rebound over.
The increasingly busy Sirigu tipped a 74th-minute effort from substitute Christian Atsu over the bar and another shot from the same player round the post four minutes later as Porto pressed for an opener.
And the decisive breakthrough the home side deserved finally came in the 83rd minute when Rodriguez curled a left-footed shot past Sirigu from 12 yards to leave the Portuguese side in a strong position to progress from Group A.