The Proteas, who finished on 213-2, closed to within 69 runs of victory thanks to a superb unbeaten hundred from opener Aiden Markram after Australia had earlier set a target of 282 runs for victory.
Markram, alongside captain Temba Bavuma (65 not out), put on an unbroken stand of 143 for the third wicket which broke the back of what would be the joint second-highest successful run chase at the Home of Cricket.
And although Australia now face an uphill challenge on day four, Vettori knows the players will take their attempted fightback one step at a time and not look too far ahead into the future.
"There's no desire to get ahead of ourselves," Vettori said. "One wicket, that's the starting point. And because these two (Markram and Bavuma) are in such control, if we can get a new batsman to the crease, then that's the start of it.
"I don't think there will be any thinking about how to get eight dismissals. It will be simply about getting one and seeing what can happen from there."
Pat Cummins' side started the day on 144-8 and appeared to have gained the upper hand thanks to Mitchell Starc's gritty half-century – his 11th in the format - having been dropped by Marco Jansen on 14 the previous evening.
The fast bowler then took the two South African wickets to fall, having Ryan Rickelton caught behind for six before seeing the back of Wiaan Mulder – caught in the covers by Marnus Labuschagne for 27 – after a half-century second-wicket stand with Markram.
At 70 for two, the game was evenly poised, but the exploits of Markram and Bavuma slowly swung the pendulum in the Proteas’ favour and Vettori was quick to acknowledge the importance of their efforts.
"Under those conditions, Markram and Bavuma were exceptional," the former New Zealand spinner continued. "They were able to navigate their way through any tricky situation and then be able to put pressure back on us at the most opportune time.
"It was a pretty benign wicket and obviously conditions weren't assisting the ball, but that partnership was exceptional."
Steve Smith left the field of play during South Africa's innings with what turned out to be a compound dislocation of the little finger on his right hand after dropping Bavuma in the slips off Starc's bowling while the batter was on two.
This Proteas team have forever written their names in the folklore of South African cricket after beating Australia by five wickets at Lord's to capture the ICC World Test Championship mace.
Aiden Markram struck a superb unbeaten century while his hobbling captain Temba Bavuma was 65 not out to leave the Proteas on the cusp of ICC World Test Championship glory at stumps on the third day at Lord's.
Pat Cummins joined an elite club of Australians to reach 300 Test wickets on day two of the ICC World Test Championship final and has his sights set on many more.
The World Test Championship final between the Proteas and Australia is still very much in the balance as the bowlers continued to hold sway at stumps on the second day at Lord's.
