Atalanta will compete in Europe's elite club competition for the fifth time since 2019.
The 1-1 draw left Juve level on 63 points with Roma and Lazio in fifth and sixth respectively, with Bologna just one point behind in seventh.
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.
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.
Inter Milan set up a potential Italian Cup semifinal clash with arch-rivals Juventus after the holders saw off Atalanta 1-0 on Tuesday.
Atletico Madrid have signed Memphis Depay from LaLiga rivals Barcelona, the two clubs said on Friday, with the Netherlands forward penning a 2-1/2 year contract.
Juventus swept aside Bologna 3-0 on Sunday to ease their recent troubles as Atalanta continued their dream start to the Serie A season with a 1-0 win over Fiorentina which kept them level on points with leaders Napoli.
Atalanta won at Sassuolo on Sunday to secure their first ever Champions League spot and Inter Milan joined them while ensuring Empoli were relegated after a frantic second half in the San Siro.
Two late goals gave Lazio a 2-0 Coppa Italia final win over Atalanta at the Stadio Olimpico on Wednesday.
