Spanked 4-1 at AEK Arena on Wednesday, pressure was on the Africans to end their poor run against European oppositions and they left it late to snatch a win at the Antonis Papadopoulos Stadium.
Chikwelu's brace plus strikes from Oshinachi Ohale and Anam Imo guaranteed Thomas Dennerby's side their first win at the Cyprus Cup.
The Super Falcons got themselves in front after Chikwelu broke the deadlock with an assist from Alice Ogebe after just seven minutes into the game.
Ohale doubled the lead for the African queens in the 28th minute before Francisca Ordega set up Chikwelu for her second and Nigeria's third of the match seven minutes from half time.
In the second half, Nigeria were handed a penalty to increase their tally but the Kristianstad midfielder missed the chance to score her third of the game from the spot in the 56th minute.
With Dennerby rigging some changes, Slovaks profited from the Africans' defensive blunders when Jana Vojtekova pulled one back in the 67th minute.
The west Africans landed themselves in some of troubles as they shockingly conceded twice from the spot, with Lucia Harsanyova's brace canceling their lead in the 72nd and 76th minute respectively.
However, substitute Asisat Oshoala teed up substitute Anam Imo for the winner as Thomas Dennerby's ladies condemned the unrelenting Slovaks to their second defeat of the tournament.
With the victory, Nigeria have now accured three points from two games and will face leaders Belgium in their last Group C match at the GSZ Stadium, Larnaca on March 4.