Marco Brescianini tapped home into an open goal with two minutes remaining to snatch a point from the Stadio Olimpico in Rome, where a passionate crowd thought a big win was coming their way.
"It's been a wonderful year, we're sorry it's ending, let's hope that 2025 will be just as good," Gasperini told DAZN.
Substitute Zaniolo decided a hard-fought encounter in Sardinia by stroking home Raoul Bellanova's cut-back in the 66th minute and moving Atalanta five points clear.
Fiorentina earlier remained on Napoli's heels after their 1-0 home win over Cagliari in their first league match since midfielder Edoardo Bove's sudden collapse last weekend.
Spaniard Raya dived low to his right to beat out Mateo Retegui's spot kick in the 51st minute then sprang across his goal to claw away the forward's follow up header.
Inter Milan thrashed Atalanta 4-0 on Friday to move provisionally top of Serie A, while Torino snatched a late 1-0 win at Venezia to continue their positive start to the campaign.
Gian Piero Gasperini lavished praise on his "incredible" Atalanta players after they stunned Bayer Leverkusen to win the Europa League and claim their first ever continental honour.
Ademola Lookman scored a stunning hattrick as Atalanta ended Bayer Leverkusen's 51-match unbeaten run to win the Europa League final 3-0 on Wednesday and claim their first trophy for 61 years.
Bayer Leverkusen had won only two major trophies in their 119-year history before this season but on Wednesday the newly-crowned Bundesliga champions have the chance to claim a double when they face Atalanta in the Europa League final in Dublin.
Bayer Leverkusen stretched their season-long unbeaten run to 50 games with a 5-0 win at 10-man Bochum on Sunday.
Atalanta's passage to the Dublin final, where they will face Bayer Leverkusen, later this month was fully deserved after a dominant performance in which Marseille barely had a kick in yet another poor away display from the Ligue 1 outfit.
The Champions League will have 36 teams in the group stage from next season, four more than this term.
Atalanta are in the last four of a European competition for the first time since 1988.
"We're not mathematically out of it but the chances of us getting there are very slim," said Calzona after his first league defeat as Napoli boss
Inter Milan cruised to a 4-0 home win against Atalanta in Serie A on Wednesday, extending their lead at the top of the standings to 12 points as they edge closer to a 20th league title.
Atalanta set up an Italian Cup semi-final with Fiorentina after winning 2-1 at AC Milan on Wednesday, while Lazio won 1-0 in a fiery derby with Roma which finished with three red cards.
Atalanta beat Sassuolo 3-1 on Wednesday to qualify for the Italian Cup quarter-finals where they will meet AC Milan.
Atalanta fans making the journey from Bergamo to see their team play Austrian side Sturm Graz in the Europa League, made a stop in Maribor, Slovenia, not to visit the city's famous castle but to surprise a former player on Thursday.
Roma slid out of the Serie A top four by the slimmest of margins as they lost 3-1 to Atalanta in Bergamo on Monday.
