Arrows lost 1-0 to Bucs on Saturday afternoon, with man of the match Qalinge deciding the last-16 tie with a 33rd minute goal at Princess Magogo Stadium.
Pirates will meet Polokwane City in the quarter-finals of the competition.
Larsen told his club's website: "A typical cut-throat cup tie with both teams playing attacking football. After a very positive opening 15 minutes Pirates then bossed the remainder of the first half with Qalinge running riot.
"We only managed to carve out one clear scoring chance to Pirates' three/four chances of which they took one.
"After the break with the introduction of [Nduduzo] Sibiya we were more enterprising and created lots more chances with [Knox] Mutizwa twice coming close with two headers as well as chances from [Thomas] Chideu and [Siphelele] Magubane.
"We gave it our all in the second half but it was not to be. Another fantastic workmanlike effort from the team of which we are very proud.
"A big thank you to all our supporters who were there to support the team."
Abafana Bes'thende next visit SuperSport United in the Premiership at Lucas Moripe Stadium on Tuesday night.
Larsen's side are a point behind log leaders Baroka after nine matches.
CAF Confederation Cup finalists SuperSport have only played four league matches owing to their exploits in Africa.