Tappara slid out of the top four after back-to-back penalty shootout losses home and away against Kiekko-Espoo.
Though Rautiainen gave them the lead in the fourth minute, thoughts of going the distance again would have been prominent after Pelicans found an equaliser through Elias Vilen early in the second period.
But the visitors' determination to get the points wrapped up came to fruition late in the second, as strikes from Aapeli Raasanen, Oliver Suvanto and a second from Rautiainen put them in firm control.
Lukas Zetterberg gave the Lahti side hope in the third period, but that was quashed with 12 minutes still to play, as Rautiainen completed his hat-trick.
The 5-2 win is the Tampere side's eighth of the season, while successive losses for Pelicans keeps them hovering above the bottom two.
Elsewhere on Wednesday, it was Vaasan Sport's turn to suffer a shootout loss, as they were edged 3-2 by TPS Turku. It's a second two-point win in a row for ninth-placed TPS, while four successive defeats leaves Sport in 13th.
Hameenlinna move into the top half after they put in a dominant display in the second period on their way to winning 6-2 against IFK Helsinki, who remain in the bottom two. Jukurit are up to fifth after edging KalPa 2-1 in regulation.
On Tuesday, Assat extended their lead at the top of the table to four points thanks to Lassi Vanhatalo's winning goal in the third minute of overtime secured a 3-2 victory over SaiPa, who lost for the third match running and have moved back to sixth following tonight's results.
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