First, make sure that the extension wire is of the same type (K, J, etc) as your sensor leads. Also, be sure you match the polarities (red to red, yellow to yellow, etc).
Best way to make the extension is using a male-female pair of thermocouple connectors. These normally have small screw terminals under the plastic cover. Something along this
Although I've never done this, you could also twist the wires together and secure them with small wiring nuts.
Twisting and soldering is acceptable as far as the thermocouple is concerned. The temperatures involved should not cause any harm to the wires, and the presence of a dissimilar metal, i.e. the solder, won't cause any problems.
But the alloys in thermocouple wires are notoriously difficult to solder. I've tried a few different fluxes and nothing has worked for me with type K wire. So I would steer you away from soldering simply because of the difficulty I've experienced in finding a flux that will wet the wires.
EDIT: The classic test for the quality of the connection is to warm the splice, but not the sensing tip or the measuring end. The meter reading should not change when the temperature changes at any intermediate point along the length of the leads.