The primary way to do that is by using the <Redirect/>component. The typical way is routing on some kind of user event. Refer the below example.

class Register extends React.Component {
  state = {
    toDashboard: false,
  }
  handleSubmit = (user) => {
    saveUser(user)
      .then(() => this.setState(() => ({
        toDashboard: true
      })))
  }
  render() {
    if (this.state.toDashboard === true) {
      return <Redirect to='/dashboard' />
    }
    return (
      <div>
        <h1>Register</h1>
        <Form onSubmit={this.handleSubmit} />
      </div>
    )
  }
}

Another way to do it is by pushing a new history to props.history.

//using ES6
import React from 'react';
class App extends React.Component {
  constructor(props) {
    super(props)
    this.handleClick = this.handleClick.bind(this)
  }
  handleClick(e) {
    e.preventDefault()
    this.props.history.push('/redirected');
  }
  render() {
    return (
      <div>
        <button onClick={this.handleClick}>
          Redirect!!!
        </button>
      </div>
    )
  }
} 
export default App;

You can use withRouter method also. Please check out the below example.

import { withRouter } from 'react-router-dom' 
class YourClass extends React.Component {
  yourFunction = () => {
    doSomeAsyncAction(() =>
      this.props.history.push('/other_location')
    )
  }
 
  render() {
    return (
      <div>
        <Form onSubmit={ this.yourFunction } />
      </div>
    )
  }
} 
export default withRouter(YourClass);