Kovacic was handed his marching orders with 17 minutes of normal time remaining at Wembley after picking up two yellow cards.
The midfielder picked up his first booking in the opening 45 minutes after going in late on Granit Xhaka.
While his sending off came on 73 minutes when he went in for 50-50 with the Swiss international.
Referee Anthony Taylor was quick to show a second yellow to Kovacic, who at the time looked somewhat confused.
But replays later showed that it was a poor decision and Kovacic had every right to be furious as he trudged off to the dressing room.
And speaking in the immediate aftermath to the challenge, Hoddle took aim at Taylor for showing Kovacic a second yellow.
He told BT Sport: “To get sent off for that is ridiculous, it really is.
“To be sent off for that as a second offensive, the first one yes of course, but take a second to look at it.
“That is where every ex professional footballer knows it is not a second offensive to be sent off. “