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The former captain guided the home side to 171-4 in the 59 overs that were possible.
He withstood some excellent South Africa bowling, particularly from Vernon Philander, who took 2-17.
England gave Test debuts to three players, with Tom Westley making 25 and Dawid Malan one.
Pace bowler Toby Roland-Jones is included in a bowling attack that is without left-arm spinner Liam Dawson, who was omitted to make way for an extra batsman.
Rain arrived before lunch, during tea and finally, just after 18:00 BST.
The four-match series is level at 1-1.
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Following a heavy defeat in the second Test at Trent Bridge, England's batsmen came in for some fierce criticism for a cavalier approach that is often not suited to the longest form of the game.
Here, they all looked to apply themselves, with Cook giving a perfect demonstration of the judgement, grit and patience that has made him England's all-time leading run-scorer.
The left-hander only played the ball when necessary, often leaving on length, scoring with tickles off the pads and pushes square on the off side.
He added 52 with Essex team-mate Westley, 49 with captain Joe Root and an unbeaten 51 with Ben Stokes, who remains on 21.
With the match evenly poised, England will look to the former skipper to push on to a 31st Test century and well beyond on Friday.
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What made Cook's knock all the more impressive was the examination provided by the South Africa pace attack on a green-tinged pitch that offered just enough assistance.
Philander managed only four overs in the morning session because of a stomach upset, but still had Keaton Jennings caught at third slip for nought.
The opener has now made three ducks in nine Test innings.
Morne Morkel tormented Root after lunch, but it was Philander who struck, a wonderful delivery matched by the one-handed catch of diving wicketkeeper Quinton de Kock.
However, it was Kagiso Rabada who produced the moment of the day, a searing inswinging yorker that bowled Malan off his pads.
Right-hander Westley, given an opportunity at number three after Gary Ballance broke a finger, was at the crease in only the fourth over and looked at home in Test cricket.
Strong off the pads, he followed Cook's example to leave well outside off stump.
He fell after lunch, loosely driving at Chris Morris to be caught at third slip.
Middlesex's Malan, a left-hander, endured a more difficult day, taking 15 deliveries to get off the mark.
But he was in no way to blame for his dismissal, with Rabada's toe-crushing pace virtually unplayable.
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Ex-England captain Michael Vaughan on TMS: "Malan got an 88mph yorker, which is always difficult, but I do think he gets a bit shut off.
"That front foot can just be opened a little bit to get in a better alignment because clearly he is going to get that ball again in the second innings.
"I can't see Jennings opening in Australia this winter but I saw him in Mumbai and thought he could play.
"Clearly he is low in confidence and I don't see him walking out at Brisbane but you never say never. The second innings is clearly going to be very big for him."
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England debutant Tom Westley, speaking to the BBC: "It was the proudest moment of my life when I got my cap this morning and it was exciting to bat but equally frustrating to get out after making a start. It's been mixed emotions in that respect.
"I was looking forward to batting full stop, but it did help ease the nerves to be batting with Alastair who I know well from Essex.
"I didn't feel miles out of my depth which is pleasing and I take confidence from. I'm looking forward to having another bat and also when we bowl because there is something in the wicket."
Alastair Cook's battling 82 held England together on a rain-affected first day of the third Test against South Africa at The Oval.