Goals from Jason Denayer, Leigh Griffiths and Kris Commons had the home side on easy street when Hoops forward Anthony Stokes was red-carded in the 87th minute for lashing out at United midfielder Paul Paton.
A minute later McGowan lunged into Henderson in the corner to also see red from referee Calum Murray before Virgil van Dijk made it 4-0 to confirm a semi-final meeting with Inverness Caledonian Thistle next month.
The Norwegian boss said: "It is stupid to tackle like that on the sidelines. We need to get that out of football.
"It is a youngster standing there with a little trick.
"You can either stay or go in with both feet, as I see it. You can injure him for life.
"But I have to see it one more time. He must be frustrated, that's why he did it.
"There were some angry and disappointed players. I spoke a bit to Stokesy, he got a knock in the his mouth and was bleeding in challenge with Paton and he reacted.
"I have to see it again to see if it was a red card.
"But I was delighted with the game."
There was little surprise about the tetchiness on show.
Deila's side beat United 2-0 in the Scottish League Cup final at Hampden last weekend when Tannadice skipper Sean Dillon was red-carded, adding to the three players, two from the Tannadice side, who were sent off in the first Scottish Cup tie on Tayside which ended 1-1.
Both teams meet again in the Scottish Premiership at Celtic Park this weekend and Griffiths expects more disciplinary problems.
The Parkhead striker said: "I think that is five (six) red cards in three games so if I was a betting man I would be putting money on a sending-off on Saturday.
"I think the Dundee United players were getting a bit frustrated.
"They might be sick of the sight of us but it is all about us and we are looking forward to Saturday and three points.
"Stokesy has a cut lip so I think something happened off the ball and he reacted and shouldn't have.
"And then then Dundee United player had a lunge at Hendo.
"It was disappointing but the main thing is we are through to the next round."
United assistant manager Simon Donnelly's first thought was that McGowan's challenge was legal.
The former Celtic striker said: "I will need to see it again. I have only seen it at game time and it was a strong fair, challenge but I might be wrong.
"The Celtic reaction on the back of their player getting sent off maybe influences but I will need to look at it again.
"Was it any worse than Scott Brown's tackle two weeks ago? (in first Scottish Cup game)."
To add grist to the mill of growing madness between the two clubs, Donnelly claimed Celtic defender Efe Ambrose should have received a second yellow card for his second-half foul on United forward Nadir Ciftci.
He said: "It was a second yellow card.
"At the start of the second half we deliberately put Nadir out there to play on Ambrose, as he was on a yellow card and I thought it was another yellow card."
