Appointment of Zajec the final straw for Redknapp and Smith
Smith claimed it was "the final straw" as the pair's relationship with chairman Milan Mandaric deteriorated.
Zajec has taken over as caretaker boss and, along with coaches Kevin Bond and Joe Jordan, steered the team to victory at Bolton.
The triumph came in the wake of the sudden resignations of managerial duo Redknapp and Smith.
Speaking on Radio 5 Live's Sportsweek programme, Smith said: "There have been problems in terms of players, what the chairman perceived and what Harry perceived.
"But bringing in a director of football pushed Harry all the way. It was the straw that broke the camel's back." Asked if he thought they would still be there had Zajec not been appointed, Smith responded: "Without a shadow of a doubt."
Smith laughed off claims that Redknapp was ready to quit after beating Manchester United at Fratton Park.
"I have never seen Harry so happy," he said. "It was a massive result.
"What a great feeling it is as a manager to manage a team that beats Manchester United."Portsmouth will launch their search for a new boss when Mandaric returns from business in Dubai and once they get the Carling Cup game against Watford tomorrow at Vicarage Road out of the way.
Chief executive Peter Storrie said: "To be honest we have not even thought about a manager yet."
A goal from captain Arjan De Zeeuw gave Portsmouth their first away league win of the season.
They could have won by a wider margin but Patrick Berger twice wasted good chances when he was clear on goal.
Portsmouth coach Bond, however, was thrilled with the way the players responded after a turbulent week.
"They were fantastic and a credit to not only Harry and Jim but themselves," he said.
"I know the players were a little bit down after what happened. They were shocked and surprised. But they have been very professional against a Bolton side that make you defend."
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