With the score tied 3-3 and about a minute remaining, Tarrant emerged from a melee deep in Southern's half.
Some deft stick-work and a surprising turn of pace enabled him to pop up in open space.
He looked up and saw Jason Dungey about 60m away and on the other side of the field. With one great heavy swipe, and with his stick reversed, he swatted the ball up the field to the waiting striker.
Dungey's first touch gave Capital's goalie Bart de Vries half a chance.
He charged forward to cut the angle down and smoother the shot, but Dungey was able to lunge forward and poke the ball past him and into the net.
It was a fantastic end to a pulsating match. Southern had spent most of the first half defending but Kane Russell opened the scoring for the home side with a superb drag flick.
The visiting side answered back through Steve Crossett seconds before the halftime break.
Russell, who looked dangerous when he ran the ball up from the back, scored a magnificent solo goal.
He charged down the right-hand side of the field.
When he got to the corner he simply weaved his way in field and along the base line until he got to the goal mouth, swerved in enough to improve the angle for his shot and let rip.
You would go a long way to see a better goal but the lead was short-lived. Capital responded immediately, when Matt L'Hullier scored from a penalty corner.
Victory appeared to have slipped away when Jacob Smith blasted the ball past a defender and unsighted goalie Leon Hayward, who stuck out an arm, more in hope, than reflex.
The ball had already crashed into the backboard.
Hayward made several superb saves, though, and stopped another deep in the match but injured himself in the process.
He appeared to have dislocated his shoulder but, after medical attention, was able to play on.
Experienced striker Chris Ashton set up the equaliser.
He robbed a defender of possession with a telling tackle and unselfishly passed to the unmarked Jeremy Morris.
De Vries got a pad in the way of the shot but spun around to see the ball trickle into the goal.
Capital pushed forward looking for the winner in the final moments, only for Tarrant to set up the victory with his bullet pass.
Southern coach Dave Ross was thrilled with the performance of his side.
"I've always rated this side and I've said I think this is the strongest Southern side that we've had," he said.
"We were really disappointed after [the 3-1 loss to Canterbury in Timaru on Saturday] but it was really good to see all the boys step up."
Ross said the difference against Capital was his side made more of its scoring opportunities.
"Against Canterbury we got 21 circle penetrations but got just one corner. Our outcomes were really poor so we focused on that in today's team talk, and we were more clinical and it showed with four good goals.
"Most importantly, the guys showed good commitment and had a high work rate.
"We displayed that right across the park and that Wellington side is a really good side.
"So it is really, really pleasing that we know we can compete with the very best in the competition.
"So that is a big step up for Southern and I'm absolutely delighted with the boys."
In other games, Auckland thumped Northland 10-0 on Saturday and beat Midlands 4-3 yesterday.
North Harbour rebounded from a 3-1 loss to Midlands to beat Northland in a penalty shoot out, and Central beat Canterbury 3-0, after having lost 4-2 to Capital on Saturday.
