How to connect L298N motor driver to 5v input supply?
Note: Since all vendors (checked Mouser, Octopart, and Sparkfun) link to the L298 datasheet when listing the L298N I will assume they are reasonably equivalent.

I will assume that the "12V-35V input pin" is the V_S pin, despite that label. There are only two supply pins on the L298 with the following electrical characteristics:


V_S, Supply Voltage, pin 4, Operative Condition V_IH + 2.546V (max.: 46V)
V_SS, Logic Supply Voltage, pin 9, typical 5V (min.: 4.5V, max.: 7V)


with 


V_IH, Input High Voltage, ranging from 2.3V to V_SS


Which would explain that the circuit works at 7.2V under the assumption that the logic inputs (enable A and B as well as inputs 1 through 4) are at close to 5V high levels. And here's the trick that might make it work: as noted V_S (the supply voltage) needs to be at a level of V_IH + 2.546V while V_IH could be as low as 2.3V and 2.3V + 2.546V &lt; 5V. So driving the L298's logic inputs with a lower voltage for logic 1, i.e. closer to 2.3V high levels, could help.  

Why is this? (source):


  [..] The actual restriction is that the motor supply be somewhat greater than the logic "1" input, but that need not match the logic supply. [..]


A warning though: 


this circuit is now operating at the fringes, one might call it shady at best, intolerable at worst
saturation voltage is related to output current and other parameters, e.g. temperature, so this might work now under certain conditions and fails horribly another day 




Therefore better bets are: 


use a higher supply voltage
ditch the L298 and its high voltage drop Darlingtons and use a MOSFET H-bridge motor driver, such as the TB6612FNG, see also
ditch the L298 and use single bipolar transistors, depending on the output current




Other things to check:

Make sure V_SS is connected to the logic supply voltage (see AnT's comment). Make sure GND of the Arduino's power supply is connected to the L298's GND.