South Africa set the West Indies a target of 298 to win the match after declaring their second innings before lunch on the final day in a bid to try and bowl out their hosts and win the first of the two-Test series.
But the 25-year-old Athanaze, in his eighth Test, scored 92 as the West Indies proved more resilient than they had in their first innings and held out to tie the contest.
They were 5-201 when the rain-affected match was called off 20 minutes from close with six overs remaining.
South Africa scored briskly at the start of the last day after resuming on 0-30 with a lead of 154 runs as they looked to set an imposing target and also give themselves enough time to bowl out the Windies in their second innings.
Tristan Stubbs scored a maiden Test half century with a quick-fire 68 off 50 balls as he put all the skills accumulated as a Twenty20 player to use.
Tony de Zorzi added 45 and Aiden Markram 38 before South Africa declared on 3-173 some 20 minutes before lunch.
The home side lost their last six first innings wickets for 60 runs on Saturday, and there was concern a similar collapse was on the cards when captain Kraigg Brathwaite was dismissed three balls into the second innings.
Spinner Keshav Maharaj opened the bowling and Braithwaite tried to slog a six off his third ball, slicing it high and being caught by Stubbs at extra cover.
The Windies slumped to 2-18 when Mikyle Louis went for nine, snagged by the speed of Kagiso Rabada, before Maharaj bagged the wickets of debutant Keacy Carty (31) and Kavem Hodge (29).
South Africa needed six more wickets in the last session but Athanaze, together with former captain Jason Holder, held firm with a 65-run fifth wicket partnership.
"We tried our best to get a result in our favour. But the wicket didn't break up as much as we expected. In the end, it was a good, competitive game of cricket," said South Africa captain Temba Bavuma.
Holder was not out on 31, along with Joshua da Silva on two when time was called. Maharaj took 4-88 for a total of eight wickets in the match.
The second Test starts at the Providence Stadium in Georgetown, Guyana on Thursday.
