Despite admitting it was an anxious ending, Allardyce was extremely pleased with the way his side held on to remain 11th on the Premier League table, three points behind 10th-placed Stoke City.
"It got a little nervy towards the end when they got the goal, but it is a fantastic result for us," Allardyce told Sky Sports.
"I thought the lads handled it really well and in this league you have to take your chances when they come.
"All-round, we were defensively solid and it was a good performance on the counter-attack."
Allardyce added: "We have just taken 15 points from eight games and that is a fantastic run at this stage of the season and that is a fantastic run considering where we were, and it has got us out of trouble and into a mid-table position.
"We have just got to strive on and see how many points we can get in the last six games.
"We aren't mathematically safe, but it is going to be very difficult for the bottom three to catch us now and the important thing about today's result was that it was against one of the bottom three, and we have taken three points off them."
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