Wijnaldum and the Netherlands earned their first win of the campaign, the Reds midfielder lasting the full 90 minutes of a 4-1 win over Belarus in Rotterdam.
The Reds' No.5 struck the post late on in Rotterdam but helped his nation to move top of Group A, on four points along with France after their opening two games, because of their positive goal difference.
The Dutch face France in a monumental fixture on Monday night at the Amsterdam Arena as both sides look to strengthen their chances of qualifying for the competition's finals in Russia in two years' time.
Wijnaldum and Klavan both complete full 90 minutes
Fellow Liverpool summer signing Klavan also had an enjoyable evening as Estonia put four past Gibraltar in a comfortable Group H victory to boost their hopes of progression.
The centre-back, captain of his country, kept a clean sheet as a brace from Mattias Kait and goals from Konstantin Vassiljev and Sergei Mosnikov ensured a comfortable win in Tallinn.
Estonia next face Greece, also on home soil, on Monday night with Klavan expected to lead out his compatriots in the Finnish capital again.
Belgium duo Divock Origi and Simon Mignolet had to watch from the bench in Brussels as the Red Devils beat Bosnia-Herzegovina 4-0 thanks to an own goal and strikes from Eden Hazard, Toby Alderweireld and Romelu Lukaku.
Roberto Martinez's men travel to Gibraltar for their next game, in which they will look to rack up the goals with Origi in the frame for at least an appearance off of the bench.
Handful of Reds youngsters in action
Meanwhile, Marko Grujic played the full 90 minutes for Serbia Under-21s in their 3-1 away win over Republic of Ireland in a qualifier for the U21 European Championship, leaving them second in their group behind Italy.
Academy youngster Ovie Ejaria featured for England Under-20s, setting up one of the Three Lions' goals as they beat Germany 3-1 in Huddersfield while the promising Trent Alexander-Arnold scored twice for England Under-19s on his 18th birthday in a 3-1 win over Croatia.