Southgate, 46, is the favourite to replace Sam Allardyce after ending his four-match interim reign unbeaten with a 2-2 draw against Spain.
“Until this run, under this spotlight, and in matches under intense pressure, you’re never sure how it is going to be,” said the former England defender.
“I’ve proved that I can handle big occasions.”
Southgate stepped up from his role as England Under-21s manager following Allardyce’s departure after just one game and 67 days in charge.
The ex-Middlesbrough boss oversaw a comfortable World Cup qualifying win against minnows Malta, before his team laboured to a goalless draw in Slovenia.
He took a significant step towards being appointed full-time manager with a 3-0 win against old rivals Scotland last week, before his team squandered a 2-0 lead against Spain at Wembley on Tuesday.
“I’m pleased and I think the remit we were given has been fulfilled,” added Southgate.
Where will I park my car? – Southgate
Southgate, who is due to return to his previous role as Under-21 manager, says he does not know when the Football Association will make a decision about his future.
If he is not given the senior team job, he will instead focus on leading the Young Lions into next summer’s European Under-21 Championships in Poland.
“I’ve no idea what happens now,” he said. “I’ve got a few days at home and a load of meetings in the diary for next week.
“We’ve got an Under-21 tournament next summer and the draw is being made next month.
“I’ve no idea which office I need to go in or which space I’m going to park my car in.”
The FA is expected to clarify its position regarding Southgate within the next 24 hours.
However, there will be a formal recruitment process and former England defender Graeme Le Saux will join the selection panel.
“I think he is suited to the ebb and flow, the rhythm of international football,” Bevington told BBC Radio 4’s Today programme.
“He has worked with these players. That was a very good performance [against Spain]. There was a tactical evolution with the ball-playing of the team.
“The players want to play for him and he is suited to that job and deserves the opportunity.”