Actually, let's expand the original reply a smidgen. If the 350 watt PSU is a known quality brand, then it should be sufficient. If it is a Chiefmax, Coolmax, Powmax, etc (in other words, a huge plethora of semi-generic PSUs that are terrifically over-rated on the labels attached to them), no, you are taking a major gamble with it.
Second, the fan motor itself draws very little current. The jack on the card is for 12 V power to the card itself, for the GPU and the RAM.