Colombian striker Zapata's goals put Gian Piero Gasperini's side seventh, just four points away from fourth place and the final Champions League spot held by AC Milan.
"We're aiming for the Scudetto (league title) now," laughed Atalanta coach Gian Piero Gasperini.
Atalanta move above Roma, both on 21 points alongside three other teams, after the capital club was threw away a two-goal lead to draw 2-2 at nine-man Cagliari on Saturday.
Leaders Juventus are eight points clear of second-placed Napoli, who smashed Frosinone 4-0 on Saturday, with Inter Milan a further three points back in third after losing to the champions on Friday.
Milan host Torino later Sunday looking to close the gap on their city rivals to just one point.
"We're in the company of great teams, everyone can fight for the Europa League and maybe even for fourth place," added Gasperini.
"It's balanced league this year, I don't remember one be so tight."
Zapata opened the scoring after just two minutes when Rafael Toloi nodded on a Papu Gomez corner and the Colombian finished from close range, only for Kevin Lasagna to quickly respond for the hosts, levelling 10 minutes later.
However Zapata struck again on 62 minutes when teenage substitute Musa Barrow failed to finish Hans Hateboer's cross and the ball fell to the striker.
The 27-year-old sealed the points 10 minutes from time with his first ever hat-trick in Italy, again off an Hateboer cross.
"Zapata's enjoying a good moment," said Gasparini, after the South American also scored in last week's defeat to Napoli, who had loaned him to Udinese.
"He is in great condition and is an extra option for us."