Kano fell 3-1 to ASC Jaraaf in Senegal after taking an early lead while Rivers lost 2-1 to Futuro Kings in Equatorial Guinea.
More was expected from both sides as the 2021 African club season kicked off with Nigeria attempting to win the African equivalent of the UEFA Europa League for the first time.
Now-defunct Dolphins came closest, reaching the 2005 final only to be beaten 3-1 on aggregate by FAR Rabat of Morocco.
"Our clubs should be made of sterner stuff," national football federation president Pinnick told the Nigerian media.
"I am at a loss to understand how Pillars can take the lead in Senegal only to collapse and lose 3-1 against Jaraaf.
"I am so disappointed with Kano and Rivers. I expected much greater spirit and steel from these clubs when they represent Nigeria in Africa.
"My challenge to both teams is to win convincingly at home and qualify for the next round of the Confederation Cup."
Rivers appear to have the easier task as a 1-0 win in Port Harcourt would take them through to the last-32 round on away goals.
Coach Stanley Eguma blamed poor finishing for the defeat in Mongomo, which would have been heavier but for an 88th-minute penalty converted by Godwin Aguda.
"There was no lack of effort from the boys, but the finishing was awful. We should have been three of four goals ahead by half-time instead of drawing."
"We must rectify this problem in the second leg, otherwise we will struggle to make the next qualifying round."
Rivers have played in the Confederation Cup once before, reaching the 2017 group stage, while Futuro are debutants.
Kano are another club debuting in the competition while Jaraaf have appeared five times, reaching the play-offs twice and making three preliminary round exits.
Pillars captain and midfielder Rabiu 'Pele' Ali believes they can overcome the two-goal deficit at home, where a 2-0 win would take the Nigerians through on the away goals.
"I realise our supporters are unhappy with the result in Senegal, but I beg them to be patient and keep having faith in us," he said.
Only once before, in 2017, have both Nigerian clubs bowed out in the preliminary round with Wikki Tourists losing to Armed Forces of Sierra Leone and Ifeanyi Ubah to Al Masry of Egypt.