The win handed South Africa a vital second point in the second-round tie, giving them a 2-1 lead.
“Winning this match was key for us,” South Africa Davis Cup captain John-Laffnie de Jager said. “It gives us a lead going into tomorrow and, with Rik playing, we know that he can play at a high level.
“It was a tough match today and we knew it wasn’t going to be easy. Those guys really return well. But I was very happy with how both guys played. The combination worked well and it’s good for us that we got the win.”
South Africa came into the tie with the score level at 1-1 after Lithuania’s number one singles player, Ricardas Berankis, defeated Jean Andersen in straight sets 6-3, 7-6 (2), 6-3, in the opening rubber on Friday.
Davis cup stalwart De Voest then levelled matters after he outclassed Laurynas Grigelis 7-6 (5), 7-6 (3), 2-6, 7-6 (4).
Berankis and Grigelis had the upper hand in the first set on Saturday, troubling Australian Open runner-up Raven Klaasen and Andersen to win the opener by six games to three.
South Africa regained their composure and bounced back in the second set, with Klaasen and Andersen breaking Lithuania’s serve twice to book a 6-3 win over the visitors.
South Africa carried this momentum into the third set, winning it in 38 minutes.
“We didn’t have the ideal start and we played an okay first game,” Klaasen said after the match.
“We went down an early break but ended the set as the stronger team. We took that mindset into the second set, because we knew that we had finished that set stronger than them. For an hour of the match we were in a good position.
“Our tactics played out well and towards the end of the match we knew there was going to be a time where we were just competing for every point and fortunately things went our way.”
Lithuania fought back in the fourth set, unsettling the South Africans, who made a number of unforced errors, and forcing the set to a tie-break. But South Africa upped their game to win the tie-breaker 7-6 (7-5).
Earlier De Voest, who made his Davis Cup debut in 2002, was handed the Davis Cup Commitment Award. The award is presented by the International Tennis Federation to players who have shown long-standing dedication in representing their country in the Davis Cup.
In the reverse singles on Sunday, De Voest will play his last match for South Africa when he takes on Berankis, while Andersen takes on Grigelis in the last match of the tie.
“With all that Rik has contributed to the game, I think him winning the match will definitely be a fairytale ending,” said De Jager.
The winner will play against the winner of the Finland v Bosnia-Herzegovina clash in September for a place in Group 1.