Foxes goalkeeper Kasper Schmeichel joined in an attack and sent a header from Danny Drinkwater's cross crashing against the underside of the bar and down.
The ball appeared to be over the line before striker Chris Wood - who was credited with the goal - made sure.
The dubious goals panel will have the final say but Shakespeare said: "I don't really care who scored. They are both claiming it.
"Kasper has been looking at the video evidence. When his header came down off the bar I didn't see the linesman signal the goal so I am pleased that Chris made sure. They can fight over it in the dressing room."
After Joe Ralls 22nd-minute free kick put Yeovil in front, Leicester were left chasing the game and Shakespeare applauded the spirit of his side.
"It was a frustrating game for us but I suppose, on reflection, you have to be delighted to get a late goal in the manner that we did," he added.
"You have to give Yeovil credit for scoring the first goal and for giving us problems.
But I also give respect to our lads for the way they kept going - you saw what it meant to them at the end in the way they celebrated.
"It showed the team spirit and the team ethic. It was a hugely important goal for us and kept our unbeaten run going. We kept that drive that you need and kept going until the end."
The draw extended Leicester's unbeaten league run to 18 games - a record for them in the second tier of English football. But Burnley narrowed the gap on Leicester to three points with their win over Doncaster while QPR are 13 points behind in third place.
Yeovil boss Gary Johnson was proud of his struggling side's performance at the King Power Stadium.
The hosts missed chances and it looked as though Yeovil were going to hold on and claim the shock result of the season - until the late twist.
"I'm proud of the boys," Johnson claimed. "They can't do any more than they have done tonight.
"We didn't come here and park the bus as some teams may have done. We created opportunities for ourselves and caused them problems.
"We kept them down to frustrated passes and they didn't get their game going. That was down to us."
Johnson believes Yeovil could have sealed the tie but admits a defiant comeback was always likely when facing the high-flying league leaders.
"We could have gone 2-0 up against a side going into the Premier League, which is what they are going to do," he added.
"They are always going to put you under pressure late on and unfortunately we did not have a marker for Schmeichel. We didn't plan on him coming up and staying up and getting his header in. If he scored, credit to him.
"I was really pleased about Leicester's celebrations. It looked as though they had just got promotion. But they had got a point against Yeovil, which just shows how far we have come."