I believe the decision of whether cut the scope or work overtime is extensively discussed on the links J provided as a comment, so I'll focus on the subject question, about how to motivate the team to have them working overtime.
There are two different scenarios, and I'd say the approach needs to be based depending on it. One is when you're working with a long-term team, and another with contractors. You have far more 'flexibility' when working with contractors, but remember that depending on the way the project is conducted (i.e. how hard you'll push the team) you may have problems in the future when you need other contractors.
There are some items, however, that worth to bear in mind when dealing with your team, as follows:
- Assess the scenarios: What happens if the project is not delivered on schedule? What will happen to the team? And to you? And with the relationship with the client? You must have these questions quite clear before action. Probably the questions you'll hear will be at some level related to them. Be prepared.
- Define priorities: You must know as well, what are the project priorities in case of delays. This must be assessed with the client (it's extensively discussed in the related links).
- Be transparent: Explain the situation to the team. Schedule a meeting where everyone feeling comfortable (i.e. avoid having this meeting in the middle of the night). Put on the table (as much as you can) the drawbacks in case the project does not deliver on time.
- Make them accountable: When you need people engaged, make them feel that their success and the project success is the very same thing. Otherwise, they may not work hard enough to deliver the project.
- Overwork guarantees nothing: Working extra-hours is not a guarantee that the project will be delivered, so while gathering team's commitment, review the scope with the client and define priorities, what is a must have and what is a nice to have.
- How they'll be compensated? They're doing extra work. They expect extra compensation. Or not. But make sure you manage properly their expectations, otherwise you may burn your team to the point of having no commitment like this in the future.
A last reminder: Working extra hours is part of the job, especially in IT. Assess what you need to do, and go for it. But make sure you're part of the team (being there with them, being available in a mobile number, helping removing any blocker they may face), otherwise they'll never respect / trust you again.