ANSWER: yes smaller tubes can fill larger tyres, but they may fail quickly and unexpectedly.
I had this exact same situation. I am converting a road bike to urban, and the largest size tire I could fit was a 700x28, the road bike has 700x23.
When I got the old tires off and pulled the tube it was sized 700x23-25. I had changed the tires and reused the old tubes and thought a 10 mile(16 km) ride ought to be safe. 5.2 miles (8.3km) later I was calling my daughter for a pickup. The front tire just went flat. No bump, rock, pothole, nail...just went flat. The rear tire stayed up the entire time, but I am changing its tube too and will keep that one in reserve. I am thinking it was the valve stem.