Walt Disney Co. raised its offer to purchase most of 21st Century Fox to more than $71.3 billion in cash and stock, topping an unsolicited offer from rival Comcast Corp. and escalating the bidding war for the coveted media properties.

Disney’s new offer is far higher than its original deal, $52.4 billion in stock, and surpasses Comcast’s all-cash offer of roughly $65 billion. In addition to having the higher offer, Disney said it also has a regulatory advantage over Comcast in winning a company to help it fight back against...