This is the third production of Shakespeare’s tragedy on the Olivier stage and, even if it doesn’t erase memories of the Peter Hall version with Anthony Hopkins and Judi Dench, it is still a terrific occasion. Ralph Fiennes and Sophie Okonedo are at the top of their game as the self-deluded lovers, and director Simon Godwin proves yet again that he combines a contemporary eye with a fastidious ear for Shakespeare’s language.

I will state my reservations quickly. This is a play of 42 scenes, and the determination to give each its separate identity means you lose some of the play’s swift cinematic rhythm. Hildegard Bechtler’s inventive design heightens the distinction between Egypt and Rome: the former all tessellated opulence and sunken pools, the latter a military war-room with the latest satellite technology. The use of a revolve, however, slows the action and means that the evening runs for three and a half hours. One consequence is that, by the time we get to the adagio of Cleopatra’s end, exhaustion is setting in.

But I’ve no wish to carp when so much is first-rate. Above all, Godwin and his two stars remind us that the lovers exist in a state of intoxicated fantasy: their opening exchanges, in their mix of lust and asperity, remind us of the warring Elyot and Amanda in Coward’s Private Lives. Fiennes especially, swathed in baggy Oriental pants, looks as if he can’t wait to get back into uniform. He goes on to give us every aspect of Antony: the born soldier, the dreaming sensualist, the wry piss-taker in the scene on Pompey’s galley.

Facebook Twitter Pinterest Fiery, funny, mercurial … Sophie Okonedo. Photograph: Johan Persson

But Fiennes is at his best in charting Antony’s tragic decline. He magnificently captures Antony’s shame after the battle of Actium and has the capacity to illuminate every phrase: when he says “this pine is barked [ie stripped] that overtopped them all” he gives an unforgettable image of a man confronting his own desolation.

Okonedo matches him blow for blow by bringing out Cleopatra’s mix of ardour and irony. She seems to love Antony most when he is not actually there as if enraptured by an imagined ideal. Like all good Cleopatras, she is also quick-witted: she assumes a self-mocking majesty with the messenger who comes with news of Antony’s marriage before ducking him in the palace pool. Okonedo is fiery, funny, mercurial and, if her final scenes are taxing, the fault lies partly with Shakespeare who camouflages the play’s choppy conclusion with poetic glory.

The supporting cast is commendably strong. Tim McMullan shows that Enobarbus is himself a tragic character in his betrayal of an Antony he once adored; Tunji Kasim catches precisely Octavius Caesar’s mix of hero-worship and contempt for Antony; and Katy Stephens justifies the gender-change of Agrippa by playing her as Caesar’s crisply efficient lieutenant. Hannah Morrish movingly shows the heartbreak of Octavia, so swiftly jettisoned by Antony, although I question the idea of bringing her back as one of Caesar’s emissaries to Cleopatra. But, all cavils aside, this is a fine production that reminds us that Shakespeare’s play is death-haunted from the start and that its self-glorifying lovers exist in a dream of passion.

• At the National Theatre’s Olivier stage, London, until 19 January.