The Three Lions' manager has come in for criticism after his team finished bottom of Group 3, failing to win any of their six outings against Italy, Hungary and Germany (D3 L3).
With the World Cup in Qatar kicking off in less than two months, the pressure is on Southgate to deliver, though he did guide England to the semi-finals in Russia in 2018 before reaching the final of the rescheduled Euro 2020 tournament last year.
Heskey – who made 62 appearances for England – thinks Southgate and his players can turn it around and can take positives from their 3-3 draw with Germany at Wembley on Tuesday (AEST).
The former Liverpool and Aston Villa striker said: "I think the draw against Germany did redeem [England] in some cases.
"I thought the fight they showed was fantastic to get it back to 3-2 because they could easily have given up at 2-0 [down]. But they fought back to make it 3-2 and then 3-3, obviously.
"Football has always fascinated me [because] we put no pressure on them, and [then] they get to a semi-final and a final, then we suddenly heap a load of pressure on them and you can see what's happening."
Prior to the six-goal thriller with Germany, England had failed to score with any of their previous 62 non-penalty shots, a run of seven hours and 30 minutes without a non-penalty goal.
Southgate's men are also winless in their last six games in all competitions, their longest run since April to June 1993 (also six). It is their longest ever winless run going into a major tournament.
"When you're not doing well, they're going to justify criticism, and you've got to accept that criticism. It's how you bounce back from that," Heskey said.
"None of these have turned into bad players overnight, and the manager isn't a bad manager overnight. I think you've got to give him the opportunity to turn it around and turn the fortune of the players around as well."
One of the concerns widely discussed has been whether Southgate knows his best team before they take on Iran, the United States and Wales in Group B at the World Cup, but Heskey pointed out selection headaches mean the manager has a number of credible options available to him.
"You could put it either way," he said. "It could be alarming, or it could be a good thing that we don't know the best 11 because players are coming in.
"You've even got Ivan Toney coming into the [last] squad... [Mason] Mount's played a lot of games and then come back in and scored.
"So it's always good to keep the manager on his toes. But you really should know who your starting 11 is coming up to a major tournament."