But Ward, 24, fought back brilliantly in the fourth set, taking it 6-3 to set up a thrilling finale.
In a tense final set it was Ward who cracked first, handing Jaziri a break point in the sixth game with a double-fault and then compounding it with another error.
But Jaziri responded in kind, though, and a netted forehand brought the set back on serve.
And that was the way it stayed until the 13th game of the set when a weary Jaziri finally netted a forehand to give Ward the chance to serve for the match.
But it was not straightforward for Ward, who was roared on by an enthralled crowd, as he was forced to save a final break point before securing victory when Jaziri dumped a forehand into the net.
Ward said: "It was a very tough match, I had to fight. I didn't play my best tennis but sometimes when you're not playing as well as you can you dig deep and find something from somewhere, and in the end I came through."
A new International Tennis Federation ruling meant the fifth rubber could have been cancelled but both captains agreed to play it and Jamie Baker rounded off a 4-1 victory for Britain with a 6-1 6-4 win over Slim Hamza.
Skipper Leon Smith took heart from the result but accepted it also showed how long the road back to respectability will be for the GB team.
He said: "You've got to be very pleased with a win. It's another step in the right direction but we should be winning matches at this level.
"You can see now when you look at someone like Jaziri, it just shows the depth in the men's game. The standard's going up and up all the time so for that reason it's just good to get through the tie.
"It was a tough match but it was a good match. I'm very pleased with the way James managed to battle through. It wasn't easy and he showed a lot of heart to get through. He did a great job."
GB's clash with Luxembourg offers a route back to the Davis Cup's Euro/Africa Zone Group I, with Murray having already indicated he is likely to make himself available for the tie.
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