Klaasen and Roelofse beat the Monaco pair 6-4, 6-4, 3-6, 4-6, 6-3 in a five-set thriller to clinch the vital point.
Klaasen and Roelofse went up two sets before dropping the third and fourth to the Monaco pair, who put up a good fight, forcing the match into the fifth set. South Africa dominated Monaco in the deciding set, posting a 6-3 winning score.
"We got out of the blocks playing well and we thought it was going to be a three set match," Klaasen said after the match.
"In this format when you go down by two sets to love, it kind of takes the pressure off and you can start to be more aggressive and I believe that's what they did.
South Africa, who are fighting to regain their spot in Group One after being eliminated by Russia 5-0 in the last tie, now need only one more point to win the tie.
"Once they were down in the match and had nothing to lose they started being more aggressive and for a couple of sets they had us rattled, but we knew that if we could stick to our guns the pressure would mount and that's what happened," said Klaasen.
Earlier Nik Scholtz upset Monaco's number one singles player Benjamin Ballere 7-5, 7-6 (3), 3-6, 3-6, 6-3 in the second singles match rubber, levelling the tie 1-1.
Scholtz, took the opening set as well as the second but allowed his opponent to come back in the third and fourth set.
The final set was a tough contest with both players tied at 2-all before Scholtz turned the heat up to claim the next game, bagging a 3-2 lead.
Scholtz then broke Ballere for a 4-2 lead and then powered on to win the set with an ace.
"This was a huge win for me, I've never played Davis Cup for my country at home before, so I was very emotional after the win," Scholtz said.
"I was an underdog going into the match and as the match started I started getting more and more belief in myself.
"I went up two sets to love and that's when reality kicked in - where I realised that now I'm ahead and now expected to win for the first time.
"I got a bit tired and nervous and I've never played five sets before, so for me physically, it's a big battle won."
In the opening singles match on Friday, Monaco's Romain Arneodo clinched a hard-fought 7-5, 3-6, 7-5, 5-7, 6-4 victory over South Africa's number one Rik de Voest, to give the visitors the lead.
The battle for points continues on Sunday in the reverse singles where De Voest will be up against Benjamin Balleret while Nik Scholtz takes on Arneodo in the last match of the tie.
"Today was great Davis Cup tennis and I'm very fortunate we got two points on the board," South African captain John-Laffnie de Jager said.
"Tomorrow it's going to be a battle of the legs, I think its going to be a running contest and fortunately for us Rik had a day off and hopefully he's ready tomorrow to get the last point."