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Dzeko, who also hit a hat-trick against Viktoria Plzen in November, took the match ball home from Villarreal after a 4-0 rout in Spain to give to his daughter after taking his Europa League-leading tally to eight goals.
Also tied with Juventus striker Gonzalo Higuain at the top of Serie A on 18 league goals, and with two from the Italian Cup, the Bosnia striker now has 28 overall -- only one short of his all-time record of 29, set at Wolfsburg in 2009-2010.
"Tonight we showed what we're made of. We won 4-0 at a big Spanish club, they were a really good side," said Dzeko.
"I'm not that surprised to be scoring so many goals. It was always like that, only last season that really didn't go well for me.
"But I don't want to stop. I want to score more."
Keeping the goals coming will be crucial at the Stadio Olimpico if Roma are to thwart a robust Torino side that hit five in a 5-3 win over Pescara last week.
Torino striker Andrea Belotti struck twice last week and is only one goal behind the league's leading pair with 17.
Roma, seven points behind leaders Juventus, can ill afford to slip up.
Juventus host relegation-haunted Palermo on Friday when Massimiliano Allegri's men will likely go 10 points clear.
Former Juventus and Palermo striker, Fabrizio Miccoli, told Tuttojuve.com: "I'm sorry, but I really don't see how Palermo can put Juve in difficulty."
Napoli, in third at just two points further off the pace, will look to make amends for a 3-1 defeat at Real Madrid on Wednesday with a win at Chievo. Napoli have won the past four encounters including those at Chievo's Bentegodi Stadium.
Roma coach Luciano Spalletti had blasted Dzeko only last month for being "too soft" in front of goal. But the striker's recent purple patch has Spalletti purring with praise.
"Italian football has improved Dzeko, said Spalletti. "He's a sensitive lad and when things don't go to plan it affects him.
"His second goal was magnificent, a goal from a mature striker. He's strong, he's got character and he will become an even better player."
Inter Milan, in fourth at 15 points adrift, sit in pole position in the race for Europa League qualification.
But Atalanta, also on 45 points, Lazio (44), AC Milan (41) and Fiorentina (40) are all in hot pursuit.
Inter travel to Bologna in Sunday's early game looking to begin another long run of successes. Before a controversial defeat at Juventus a fortnight ago, Stefano Pioli's men had put seven together back to back.
Atalanta have already surpassed expectations in Bergamo and have perhaps the easiest fixture of all the Europa League chasers, at home to struggling Crotone on Saturday.
Lazio, meanwhile, face a potentially tricky fixture at Empoli less than a week after succumbing to Suso's late equaliser for Milan in a 1-1 Stadio Olimpico draw.
Coach Simone Inzaghi expects to see improvement in his front line after a total of 25 shots produced only one goal on Monday.
"If you have 25 shots, you have to win 2-0," said Inzaghi, the brother of former AC Milan striker and coach Filippo.
Spanish forward Suso, who joined Milan in 2015 after an unsuccessful three-year spell at Liverpool, is in the form of his life and will hope it continues Sunday when they look to boost their qualification chances with a win against Fiorentina.
"We will do everything we can to try and qualify for the Europa League," he said.
Fixtures (times GMT)
Juventus v Palermo (1945)
Atalanta v Crotone (1700), Empoli v Lazio (1945)
Bologna v Inter Milan (1130), Chievo v Napoli, Pescara v Genoa, Sampdoria v Cagliari, Udinese v Sassuolo (all 1400), Roma v Torino (1700), AC Milan v Fiorentina (1945)