The hosts looked to be in a precarious position on 121-6 in their second innings, after bowling out Sri Lanka for 453 to claim a lead of 122.
However, Ballance (104 not out) led the fightback with a hundred in only his second Test, offering England hope that they may have found some security at the troublesome number three spot.
With assistance from Chris Jordan (35) and Stuart Broad (24), Ballance helped his side to 267-8 and an advantage of 389, bringing up his century in style with a six during the final over of the day.
An overnight declaration from England appears likely, with Sri Lanka already needing to complete a record run-chase at Lord’s in order to claim victory.
Having started day four on 415-7, Sri Lanka lost their final three wickets for just 38 runs.
Number 11 Nuwan Pradeep (4) departed in spectacular fashion to bring the innings to an end, crashing into his own stumps after being struck by a short ball from Chris Jordan (3-102).
Yet Sri Lanka responded superbly, thanks largely to the efforts of paceman Shaminda Eranga (3-63).
Captain Cook and Test debutant Sam Robson initially looked well set, but England collapsed after the former was caught behind off Eranga for 28.
Robson (19) duly played on facing the same bowler and Eranga also bowled Ian Bell (9) during a wonderful spell.
Joe Root (15) – a double-centurion in England’s first innings – was then trapped lbw by Rangana Herath (4-95), who struck again with a wonderful, flighted delivery to bowl Moeen Ali through the gate for four.
And Sri Lanka will have been confident of dismissing England for a relatively modest total when Matt Prior (16) was caught in the gully off Nuwan Kulasekara (1-65).
However, Ballance held the innings together with a gritty knock and finally found able support from Jordan and Broad.
Although both lower-order batsmen were dismissed before the close, there was time for Ballance to reach three figures, with a slog-swept six off Herath rounding off an excellent period of acceleration as England sought to push on prior to their anticipated declaration.
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