<script src="http://unpkg.com/vue/dist/vue.js"></script>
<script src="http://unpkg.com/element-ui@2.3.8/lib/index.js"></script>
<div id="app">
<template>
 <div class="block">
  <span class="demonstration">起始日期</span>
  <el-date-picker v-model="startDate" type="date" placeholder="選擇日期" :picker-options="pickerOptionsStart" @change="changeEnd">
  </el-date-picker>
 </div>
 
 <div class="block">
  <span class="demonstration">截止日期</span>
  <el-date-picker v-model="endDate" type="date" placeholder="選擇日期" :picker-options="pickerOptionsEnd" @change="changeStart">
  </el-date-picker>
 </div>
</template>
</div>

var Main = {
  data() {
   return {
    pickerOptionsStart: {},
    pickerOptionsEnd:{},
    startDate: '',
    endDate: '',
   };
  },
  methods:{
   changeStart (){
    this.pickerOptionsStart = Object.assign({},this.pickerOptionsStart,{
     disabledDate: (time) => {
      return time.getTime() > this.endDate
     }
    })
   },
   changeEnd (){
    this.pickerOptionsEnd = Object.assign({},this.pickerOptionsEnd,{
     disabledDate: (time) => {
      return time.getTime() < this.startDate
      }
    })
   }
  }
 };
var Ctor = Vue.extend(Main)
new Ctor().$mount('#app')

@import url("http://unpkg.com/element-ui@2.3.8/lib/theme-chalk/index.css");

.block{
 margin-top:10px;
}