Connect device to power
... an electronic device ... should be powered by 10 - 24 V 300mA max
  I have a power supply which outputs 12 V 1.1 A
  Will the supply power the device correctly?


Yes, probably.

(1) Power supply voltage must be inside the range specified 

As 10 V minimum wanted  &lt; 12V available &lt; 24V max wanted the voltage is OK.

(2) Power supply current is what the supply CAN supply.
 As long as the demand by the equipment is not greater than the capability it is OK.

Here 300 mA wanted &lt; 1.1A capability so it is OK.

Note:


Polarity must be correct.
Supply must be DC if device wants Dc and AC if device wants AC.
Some supplies have more 'ripple' than others and mat not work well with some equipment but your equipment is probably non demanding.
Some manufacturers use special circuitry to stop their equipment working with other than their own power supplies. Not usually a problem.
eg Some Dell equipment.
Early Razr cellphones.