Nathan Cotterrall and Kaid Mohamed had early chances in the 30-degree heat, but Antti Hynynen struck first.
Ville Lehtinen added a second and Jussi Pekka Savolainen a third at the break.
An own-goal header from Colin Loss made it four, Aristides Pertot's late goal seeming to make the 24 June return at Newtown's Latham Park a formality.
Carmarthen manager Mark Jones:
"We're obviously disappointed and the second leg is probably dead now, but we had our chances.
"We can't complain about the heat - we should be more used to it than them, they're Finns!
"An awful 20 minutes really cost us. If we want to be giant killers we can't give away own goals and play like that.
"I could have accepted 2-0, but any more than that was unflattering.
"They were clinical and showed why they're top of the league, but I really think we can win the second leg if we pick our heads up."
Tampere United: Mikko Kavén, Vasile Marchis, Jussi Kuoppala, Antti Hynynen (Antti Ojanperä 86), Ville Lehtinen (Miki Sipiläinen 66), Jussi-Pekka Savolainen, Sakari Saarinen (Juska Savolainen 69), Jarkko Wiss, Petri Heinänen, Heikki Aho, Aristides Pertot.
Subs not used: Mika Hilander, Erik Westerholm.
Yellow cards: Lehtinen 52, Sipiläinen 85.
Carmarthen: Mark Ovendale, Luke Hardy, Martyn Giles, Neil Smothers, Kevin Aherene-Evans, Rhodri Jones (Sacha Walters 63), Chris Thomas, Colin Loss, Mark Dodds (Craig Lima 83), Kaid Mohamed (Mattie Davies 63), Nathan Cotterrall.
Subs not used: Andrew Delve, Wayne Jones.
Yellow cards: Loss 59.
Referee: Nebojsa Rabrenovic (Serbia & Montenegro)