"When the players are good, every system works," Ranieri said, deflecting the praise away from him. "The most important thing is the team."
Bastia started brightly and winger Wahbi Khazri tested goalkeeper Danijel Subasic with a low show, before Monaco hit back and goalie Mickael Landreau made a superb save to deny Riviere.
"We didn't play with fluidity for the first 25 minutes, so there's still work to be done," Ranieri said.
Riviere broke the stalemate in the 39th and Falcao made it 2-0 moments later.
Riviere is quick, and given that he had been rested by coach Claudio Ranieri for the past two matches, looked full of energy. Colombia winger James Rodriguez picked him out with a long pass and he held off defender Francois-Joseph Modesto before firing home.
Joao Moutinho's pass for Falcao was even better, and he beat center back Sebastien Squillaci for pace before shooting across Landreau. He got his second in the 89th, finishing confidently after good work from winger Yannick Ferreira Carrasco.
Although Valenciennes has the second worst defense in the league, PSG struggled to get on top and spent large portions of the match on the back foot.
"It was a difficult match in every sense of the term, but the main thing is to get three points because we'll have a lot of games like this during the season," PSG coach Laurent Blanc said. "We played with determination but we were very clumsy at times."
Despite its expensive collection of stars, PSG struggles to play with any fluidity and is heavily reliant on individual skill to break down teams.
PSG is without first-choice center half Thiago Silva for the next six weeks due to a hamstring injury and used Brazilian center half Alex alongside 19-year-old Marquinhos. The makeshift pairing was tested throughout.
Valenciennes gained in confidence during the first half.
As PSG's frustration grew, winger Ezequiel Lavezzi got a yellow card for arguing with the referee; Marco Verratti was booked for a blatant foul on midfield counterpart Mathieu Dossevi, and so was Digne for a thudding challenge on Rudy Mater.
But moments before the halftime whistle, Cavani finished off some fine work from Ibrahimovic with a strike into the bottom corner for his third league goal of the campaign.
"The Cavani-Ibra partnership is getting better," Blanc said, adding that he is relieved Ibrahimovic has signed a one year extension to his contract until 2016. "Tonight he made the difference with a bit of class."
Verratti cleared defender Lindsay Rose's header off the line moments after the interval, Israel striker Maor Melikson's shot was blocked by Marquinhos, substitute Anthony Le Tallec curled a free kick just wide, and goalkeeper Salvatore Sirigu stopped Dossevi's deflected effort in injury time as PSG just held on.
In other results, it was: Ajaccio 2, Lyon 1; Bordeaux 0, Reims 0; Guingamp 5, Sochaux 1; Nantes 2, Nice 0; Toulouse 1, Lorient 0.
Nantes moved up to ninth place with goals from Serbia striker Filip Djordjevic and Fernando Aristeguieta, while Denmark forward Martin Braithwaite grabbed a late winner for Toulouse.
Ajaccio's win moved it out of the relegation zone, while Bordeaux dropped down into 18th place.
Montpellier hosts Rennes on Thursday night.
On Tuesday, Marseille beat Saint-Etienne 2-1 and Lille beat Evian 3-0.
