Emily Thornberry said Labour would not "frustrate Brexit" even if it failed to amend the bill.
Ten shadow ministers were among 47 Labour MPs who rejected party orders to back it last week.
Shadow chancellor John McDonnell said frontbenchers would have to quit if they defied the whip at the next vote.
The draft legislation comes back to the Commons on Monday for three days of debate culminating in a vote on its third reading.
Speaking on BBC's The Andrew Marr Show, Ms Thornberry rejected a suggestion the party was "hopelessly divided" on the issue.
She said she understood colleagues not wanting to vote in favour of invoking Article 50 of the Lisbon Treaty, which begins the formal departure process, but said: "We have said that we will not frustrate Brexit. We have got our instructions from the British people. We are democrats and the public have voted to leave the European Union."
Labour will try to amend the draft legislation to secure what it calls a "meaningful vote" on the final deal struck between Theresa May and the EU, and to guarantee the status of EU nationals in the UK and Britons living elsewhere in Europe.
Shadow business secretary Clive Lewis has said he will vote against the bill unless Labour's amendments are accepted, describing them as "red lines".
Ms Thornberry declined to say whether they were non-negotiable red lines for her party, and denied it was "illogical" to demand amendments but still back the bill in the final vote if they are rejected.
"There will need to be back channels, private conversations. There are many conversations going on now," she said.
"We are speaking to government, we are speaking to Tory backbenchers and we are trying to get a compromise that will work."
One of Labour leader Jeremy Corbyn's closest allies, shadow home secretary Diane Abbott, missed Wednesday's vote, citing illness.
Former Labour minister Caroline Flint criticised Ms Abbott, telling ITV's Peston on Sunday: "We used to have man flu, we now have Brexit flu that Diane has created hereâ€¦ it's about being part of a team.
"I think she holds one of the most important portfolios within a shadow cabinet. If she can't support the leader on this then she should go."
But former Labour deputy leader Harriet Harman defended Ms Abbott, pointing out she had earlier said on television she would back the bill so had "already taken the flak on it".
Among the rebels in last week's second reading vote were 10 members of the front bench, who would ordinarily be expected to step down after defying leadership instructions to back the bill.
Labour leader Jeremy Corbyn told BBC Radio 4's The World This Weekend that changes to his shadow cabinet would be announced "in the coming few days".
Asked if he should be lenient, he added: "I'm a very lenient person."
Speaking later, he said he expected Ms Abbott to vote with the party this week.
Shadow chancellor John McDonnell said the shadow cabinet would decide how to approach Wednesday's third reading vote on the Brexit bill.
He told Pienaar's Politics on BBC Radio 5 live that if a three-line whip was again imposed, any frontbenchers who rebelled "will have to step down".
The shadow foreign secretary has suggested Labour will continue to support legislation paving the way for Brexit as it passes through Parliament.