England were 138 for six in their second innings, a lead of 274 runs, at tea on the third day.
Jonny Bairstow, fresh from his first-innings 99, was nought not out after 23 balls faced, with Moeen Ali unbeaten on four.
Pace bowler Olivier, only called up for this match after back injuries ruled out both Vernon Philander and Chris Morris, had taken two wickets for 13 runs in eight overs.
England were bidding for a first home Test series win over South Africa since 1998 at 2-1 up with one to finish in this four-match contest.
Root was also in search of a victory in his first Test series as England skipper.
But after establishing a healthy first-innings lead of 136, England lost three wickets either side of lunch.
After Keaton Jennings had been dropped on nought by diving third slip Dean Elgar off Kagiso Rabada, fast bowler Morne Morkel removed Alastair Cook and Tom Westley.
Cook's Essex team-mate Westley then managed just nine in his fourth innings at this level before he too was caught in the gully.
The towering Morkel, who'd so often bowled well this series without reward, had taken two for eight in 18 balls.
- Jennings fails again -
========================
At lunch, England were 53 for two, with Jennings 17 not out and Root 13 not out.
But South Africa-born Jennings was unable to cash in on his reprieve.
Prior to today's innings, there had been widespread speculation that the 25-year-old would be dropped for the start of England's upcoming three-Test series at home to the West Indies, which precedes an Ashes tour of Australia.
And Jennings -- the 11th batsman chosen as an England opening partner for Cook since former captain Andrew Strauss retired in 2012 -- moved close to being axed after totalling just 127 runs in four Tests this series at an average of 15.88.
Dawid Malan, out for scores of one and 10 on Test debut during England's 239-run victory in the third Test at The Oval, was looking for a big score after falling for 18 in the first innings of this match.
But the Middlesex left-hander, born in London but brought up in South Africa, exited for six -- having been dropped on five -- when he gave a simple bat and pad catch to short leg off left-arm spinner Keshav Maharaj.
But Root and Stokes added 57 valuable runs.
Root had made a composed 49, featuring six fours, when he played on to Olivier.
Olivier struck again when a ball angled across left- handed batsman Stokes, on 23, took an outside edge held by South Africa captain Faf du Plessis in the slips.
Earlier, Stuart Broad removed Olivier as the, nine down overnight, were dismissed for 226.
It meant James Anderson, England's all-time leading wicket-taker -- was denied a maiden five-wicket Test innings haul on his Lancashire home ground.
But figures of four for 38 in 17 overs meant Anderson, who'd struck four times from the newly-named James Anderson End on Saturday, had his best innings return in seven Tests at Old Trafford.
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