Wijnaldum followed up a tide-turning performance that saw him score one and assist one in a 4-2 victory over Germany with another impressive display as the Dutch won in Tallinn.
Liverpool’s No. 5 started in midfield again, this time joined by Davy Propper alongside Frenkie de Jong, while Virgil van Dijk was partnered with Matthijs de Ligt at the back.
The Dutch opened the scoring as a brilliant, low cross from Daley Blind was touched home by Ryan Babel, with the ex-Red in inspired form.
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Babel doubled the lead with a header just after half-time, with Van Dijk almost netting one of his own as he rose highest to nod a corner just wide of the post.
The best goal of the night game from Memphis Depay, touching down a long ball from De Ligt and, cutting onto his left, placing a perfect effort beyond Sergei Lepmets from outside the box.
Depay then laid on the fourth in the 87th minute, steering a free-kick into the area for Wijnaldum, leaping in front of Van Dijk, to head into the back of the net via some questionable goalkeeping from Lepmets.
The 4-0 win pushes the Netherlands to within three points of Germany and Northern Ireland at the top of Group C in the Euro 2020 qualifiers, having played one game fewer.
Only Depay (four) and Serge Gnabry (five) have scored more goals than Wijnaldum (three) in Group C so far, with the 28-year-old taking to Twitter after the game to celebrate another victory:
In the tunnel after the game at Lillekula Stadium, there was an opportunity for old team-mates to swap shirts, as the two captains, Van Dijk and Klavan, embraced:
— OnsOranje (@OnsOranje) September 9, 2019
Klavan left Liverpool just six months after Van Dijk’s £75 million arrival from Southampton, with the much-loved Estonian now plying his trade for Cagliari in Serie A.
Achilles problems blighted his first season in Sardinia, but Klavan has been an ever-present in Cagliari’s first three games of 2019/20 so far, though they currently sit 18th in Serie A.