Cian Ashford's early double either side of a Salvatore Salech strike put Cardiff 3-0 up inside 39 minutes, with Roma's defence still short of match sharpness and Paulo Dybala missing a penalty before eventually pulling one back himself in the second half.
Tanner restored Cardiff's three-goal cushion soon after, and the afternoon was further soured by a spat between Bryan Cristante and Matias Soulé that had to be broken up by team-mates. Speaking afterwards, Gasperini turned his attention to the transfer market rather than dwelling on the result, reiterating that his squad remains incomplete following a busy summer of departures.
The head coach was clear about what he still needs: a defender, a wide player and two forwards, plus any further reinforcements required should more players leave, stressing that a side competing in the Champions League needs at least two reliable options in every position to rotate without losing quality.
On new signing Castro, who made his debut as the central striker against Cardiff, Gasperini was full of praise despite a quiet individual performance.
"It's certainly an excellent signing, it was a great piece of business. These are the kind of players who improve you. Essentially Castro is a centre-forward and (Donyell) Malen, for me, is a centre-forward too. Then there are situations where they can play together.
"We have the prospect of playing a huge number of matches. We saw last year, in the second half of the season, that at a certain point we used Malen in every single game without ever giving him a break. If that happened again this season, we would risk losing part of his quality.
"When you're playing Champions League football at a certain level you need a complete, competitive squad with enough players who can rotate without losing quality of the change. Castro is a 2004-born lad who wanted to come to Roma no matter what, so he'll be a good signing," Gasperini said.
Castro joined from Bologna for €35m including bonuses, with Roma agreeing to hand the Emilian club a 10 per cent cut of any future sale.
AS Roma also confirmed the signing of centre-back Konstantinos Koulierakis from VfL Wolfsburg on the same day, in a permanent deal worth around €17m. The 22-year-old Greek international, who previously played for PAOK before his move to Germany, has signed through to 2031 and will wear the number 33 shirt.
"AS Roma is delighted to announce the arrival of Konstantinos Koulierakis from Wolfsburg... Welcome to Rome, Konstantinos!" the club said in a statement.
Standing close to 1.90m, left-footed and known for his pace and aggression, Koulierakis made 59 appearances and scored four goals across two Bundesliga seasons and has 22 caps for Greece.
Inter had previously made enquiries about him, while Juventus considered a move last summer in case of a Kelly departure, though it is Roma who have now completed the signing as Gasperini looks to shore up a defence that struggled badly in Cardiff.