With the tie level at 2-2, US Open finalist Pliskova teamed up with Strycova to defeat French Open doubles champions Caroline Garcia and Kristina Mladenovic 7-5, 7-5 to prolong the Czechs' reign.
It was their third triumph in succession and 10th overall, with only the United States boasting more titles with 17.
"Unbelievable. We don't have words yet, we are going to celebrate. The feeling is something so special, even if it's our fifth title. It's really unbelievable," said Strycova, a member of the title-winning teams in 2011, 2012, 2014 and 2015.
France had grabbed a 2-1 lead earlier in the day after Garcia moved them within reach of a first Fed Cup crown since 2003 following a 6-3, 3-6, 6-3 win over sixth-ranked Pliskova.
But Strycova, drafted in to replace injured two-time Wimbledon champion Petra Kvitova by Czech captain Petr Pala, kept their hopes alive by overcoming Alize Cornet 6-2, 7-6 (7/4) to force a fifth and final rubber.
Mladenovic sat out the reverse singles complaining of cramp after losing a four-hour marathon to Pliskova in the opening match on Saturday, but returned to partner Garcia in the doubles.
Pliskova and Strycova then broke again to move 3-2 ahead in the second, but the French pair responded to draw level at 4-all.
Mladenovic's serve came unstuck once more though when Pliskova reacted superbly to dig out a volley and convert the decisive break point for a 6-5 lead as the Czech Republic extended their recent domination.
France had been seeking to win a third crown -- the last of which came when captain Amelie Mauresmo inspired them to a 4-1 triumph over the United States 13 years ago.
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