Gabriel Sakellaridis was speaking after Mr Tsipras won the late-night vote convincingly, but with more than 30 of his own MPs voting against him.
The vote means the government lost its majority but is expected to survive.
Eurozone finance ministers are due to discuss the vote in a conference call.
On Wednesday the European Commission proposed a €7bn (£5bn) "bridging" loan to help Greece pay debt interest due in several days.
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Under the terms of the bailout deal agreed in Brussels on Monday, the first tranche of legislation relating to tax and pensions had to be passed by Wednesday.
The Greek parliament discussed the bailout deal beyond a midnight deadline into Thursday morning.
Mr Tsipras won the vote by 229 votes to 64, with the support of some opposition MPs.
See how events unfolded here
The BBC's Mark Lowen in Athens says he has been weakened and will now need a reshuffle or a vote of confidence.
Passionate opposition came from his own party, with parliamentary Speaker Zoe Constantopoulo calling the measures "social genocide".
Former finance minister Yanis Varoufakis was another vocal opponent of the measures.
Energy Minister Panagiotis Lafazanis and Deputy Labour Minister Dimitris Stratoulis also voted against the package, prompting speculation they could lose their jobs in a reshuffle.
In his address to parliament Mr Tsipras said: "I acknowledge the fiscal measures are harsh, that they won't benefit the Greek economy, but I'm forced to accept them."
He added: "The Greek people can understand the difference between those who fight in an unfair battle and those who just hand in their weapons."
He also said it was a "positive development" that one of Greece's creditors, the International Monetary Fund (IMF), had suggested that the bailout does not go far enough - and that some debts will need to be written off.
Mr Sakellaridis said the vote was the first important step towards a deal and hinted that the government would try to remain in office despite losing its majority.
"The basic priority of the prime minister and the government is the immediate and successful completion of the agreement," he said.
When Eurozone finance ministers hold their conference call at 08:00 GMT they will discuss the next steps.
Later, the European Central Bank's governing council will meet to discuss how to keep Greece's banks from collapsing.
Greek banks have been closed for more than two weeks to reopen, with cash withdrawals limited to €60 a day.
For Greece to secure its new funds, it must win the approval of the domestic parliaments of several of the other 18 members of the eurozone. Germany is expected to vote on the deal on Friday.
Greece must also commit to a major overhaul of the civil justice system by 22 July and agree to more privatisation, to review collective bargaining and industrial action and to make market reforms, including Sunday trading.
Thursday morning's vote approved:
Opponents of the bailout measures took to the streets of Athens in mainly peaceful protests ahead of the vote. One group threw petrol bombs at police officers who responded with tear gas.
Unions and trade associations representing civil servants, municipal workers and pharmacy owners also went on strike on Wednesday.
Greek PM Alexis Tsipras is focused on completing a €86bn bailout deal agreed with the eurozone despite setbacks in a crucial vote to push through tough reforms, his spokesman has said.