Wijnaldum's first-half header set the Reds on the road to victory at Wembley and Firmino's second-half tap-in after a rebound off the post ultimately secured the three points.
Liverpool's attacking exertions were matched by defensive grit in the latter stages of the match, as Spurs battled back and Erik Lamela snatched a consolation goal in injury-time.
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Gini's away day
Wijnaldum's 39th-minute opener at Wembley was the Dutchman's first away goal in the Premier League in 58 appearances on the road in the English top flight.
It arrived after Spurs goalkeeper Michel Vorm pushed away a corner and was then unable to claw out the midfielder's header from under the bar, with the Goal Decision System instantly indicating the ball had gone in.