The Bears were charged with misconduct by tournament organisers European Professional Club Rugby after picking lock Elliott Stooke as a replacement in their match-day squads for both fixtures last month.
An independent disciplinary committee heard submissions from senior Bristol officials, with the club accepting they had breached tournament rules.
The points deduction drops Bristol to fifth in their Challenge Cup pool, but they have still qualified for the round of 16.
A 10,000 euros (£8,700 fine), meanwhile, has been suspended until the end of next season. Bristol say they will not be appealing against the decision.
In a statement, EPCR said: “The committee heard that Stooke, who had been previously made redundant by Wasps, was properly registered to play for Bristol Bears as an additional player in the EPCR Challenge up, and that under EPCR’s rules, additional players must be contracted to the club for a minimum period of three months.
“In addition, the committee heard that clubs must also sign an Additional Player Undertaking to ensure that the three-month contract remains in full force and effect for the entire three-month period, and to ensure that players do not enter into a contract or arrangement with any other club that would prevent them from playing for the club with which they have been registered.
“The committee was told that following his participation in the matches against USAP (Perpignan) and Zebre Parma, Stooke informed Bristol Bears that he was availing of an early release clause in his contract and was taking up an offer to join Montpellier.
The member of staff responsible had made an honest clerical mistake when registering him as an additional player
European Professional Club Rugby
“However, by permitting Stooke to sign for Montpellier before the three-month period had elapsed, Bristol Bears unwittingly breached the Additional Player Undertaking, and Stooke therefore became retrospectively ineligible for the round one and two matches.
“The committee heard that as all relevant members of Bristol Bears’ staff were not aware that Stooke had an early release clause in his contract, the member of staff responsible had made an honest clerical mistake when registering him as an additional player.
“While accepting that Bristol Bears had committed a genuine error and had not sought to gain an unfair advantage in any way, the committee determined that with the need to uphold the integrity of the EPCR Challenge Cup as paramount, the club was guilty of misconduct due to a breach of EPCR’s tournament rules.”