Real Madrid, the 2011 winners, fell two goals behind at La Liga rivals Celta Vigo but Cristiano Ronaldo's late strike to make it 2-1 put the 2011 winners in a reasonably strong position ahead of next week's second leg at their Bernabeu stadium.
Messi's Argentine compatriot, Athletic Bilbao coach Marcelo Bielsa, had a wretched night when his side were dumped out by third tier Basque neighbours Eibar on away goals in their postponed last-32 second leg.
Mikel Arruabarrena's 72nd-minute penalty for Eibar at the San Mames sent the Segunda B side through after the second leg ended 1-1.
Eibar, who held Bilbao to a 0-0 draw in the first leg, will play Malaga in the last 16, with a quarter-final against Barca or Cordoba the prize for the winners.
On the other side of the draw, Atletico Madrid have one foot in the last eight after forward Diego Costa scored twice in a 3-0 win at home to city rivals Getafe.
Atletico, resurgent under Argentine coach Diego Simeone, would play second division Las Palmas or Real Betis in the quarter-finals, with the pair meeting on Thursday.
On Sunday, World Player of the Year Messi overhauled Gerd Mueller's goals record for a calendar year of 85 set in 1972 and he took only 11 minutes on Wednesday to add to his haul in a vibrant atmosphere at Cordoba's packed Arcangel stadium.
David Villa sped clear on the left and he squared for Messi to clip the ball into the net off the underside of the crossbar from close range with the Cordoba goalkeeper stranded.
Fielding a near full-strength team despite the lowly opposition, Barca controlled the game in the second half before Thiago Alcantara released Messi on the left in the 74th minute and he finished low into the corner.
Barca coach Tito Vilanova said Messi's injury scare in last week's Champions League match against Benfica, when he bruised a knee in a clash with goalkeeper Artur, had sharpened the 25-year-old's hunger for goals.
"I think he's so happy after the fright against Benfica that now he's even more motivated," Vilanova, whose side are six points clear at the top of La Liga and through to the last 16 of Europe's elite club competition, told a news conference.
Bruno Lucena, head of Flamengo's research and statistics department, told Brazilian sports daily Lance that he had calculated Zico scored 89 goals in 1979 and Messi's record is not valid. The statement could not be verified.
WILD CHALLENGES
Real struggled against a well-drilled Celta side in rain-lashed Galicia and fell behind when Mario Bermejo fired between reserve goalkeeper Antonio Adan's legs from a narrow angle in the 54th minute.
Cristian Bustos made it 2-0 moments after coming off the bench in the 78th with a superb long-range effort that arrowed into the top corner before Ronaldo, who was lucky to be on the pitch after throwing himself into a couple of wild challenges, pulled a goal back three minutes from time.
"It was a tough ga
