The Centrica-owned supplier left tens of thousands of business customers with delayed or inaccurate bills after it launched a new IT billing system in March 2014, according to regulator Ofgem.
More than 6,000 new customers had delays in registering with the supplier, and British Gas Business handled many complaints over the issues "poorly or not on time".
Ofgem said British Gas Business came forward voluntarily in December 2014 to flag up the billing issues.
Dermot Nolan, chief executive of Ofgem, said that while the firm was right to report the problems, it "didn't act quickly enough to put things right".
He added: "Many existing business customers were left without accurate bills, causing significant concern.
"New customers were delayed from switching to the supplier, potentially costing them money by leaving them on a more expensive deal with another supplier."
Ofgem said British Gas Business failed to protect customers from issues surrounding the launch of its new IT system, did not ensure its customer service processes were adequate and then did not have enough staff to deal with complaints.
British Gas Business, which has around 400,000 customers, initially compensated some affected customers after discovering the billing issues, took action to cut the number of delayed bills and contacted customers to explain problems with the billing system.
It allowed some customers to switch despite being in debt, if the debt was related to its failure to produce a bill.
The provider has since improved its service and hired a team of up to 350 specialist agents to help handle the increased number of calls and ongoing complaints.
Mr Nolan said: "Ofgem welcomes suppliers investing in new systems and recognises that this can be challenging.
"However the £9.5 million payment also sends a strong reminder to all energy companies that they must treat consumers fairly at all times, including while new systems are put in place."
The fine includes compensation to customers impacted by the failures, as well as payments to the Money Advice Trust charity, which provides the Business Debtline service.
British Gas Business apologised to those affected and said its billing issues are resolved.
Gab Barbaro, managing director of British Gas Business, said: "We invested in a new billing system so we could improve the service we provide to our business customers. At the time, this was a major undertaking - merging nearly 100 different systems into one.
"It didn't go as smoothly as we would have liked so we reported this to Ofgem as a priority.
"I'd like to apologise to our business customers for any inconvenience caused when we were implementing the system."