The Light Blues required a stoppage-time goal from Harry Forrester to give them a 1-0 win over Dundee last week but looked to be heading towards their first defeat by the Jags since 1993 after Kris Doolan scored in the 76th minute.
However, Dodoo, on for Joe Garner on the hour mark, levelled with a finely-executed volley in the 81st minute before drilling in the winner in the fourth of four added minutes.
It was enough to take the Ibrox side above Aberdeen into second spot in the Scottish Premiership by two points.
“I thought he did well,” said Warburton. “He started central, we have used him out wide but we have striking options and he has banged on the door and said he wants his chance, and all credit to him.
“He is a great young professional, he is first in every morning with the masseur, he is always working on his finishing and I am delighted for him.
“He was asked to make a difference when he came on and he did that.”
St Johnstone manager Tommy Wright has accused Neil McCann of deflection tactics as he denied reporting the Dundee boss to the Scottish Football Association.