Nolan struck towards the end of the first half in typically predatory fashion to give the newly-promoted Hammers three points over Newcastle and move them up to sixth place in the Premier League table.
Speaking in his post-match press conference, the 58-year-old hailed the former Newcastle duo's almost telepathic understanding on the pitch and its huge benefit to his side's chances of success.
"I think that Kevin's relationship with Andy is such a good one that I don't need to do a lot of coaching with them two," Allardyce told reporters.
"Kevin will instinctively play off Andy and vice-versa. While Andy's struggling to get a goal, and we'd like him to get one, Kevin's the one scoring the goals because that combination is causing teams lots of problems."
The victory at St James' Park leaves West Ham with 18 points from 11 matches played, only two off Everton and the fourth Champions League place, a tally that Allardyce believes will provide his side with a good platform for the challenges of the season ahead.
"It's a fantastic start to the season for us now," he continued.
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"That's the pleasing thing for me because confidence grows and grows and grows as you could see today. It was a very professional performance from the players, nobody was overawed coming here."
The former Bolton and Blackburn manager was also quick to praise the way his side kept the ball amidst common perceptions of their apparent tendency to use the long ball.
He added: "I thought our tactical approach today was spot on. Our in-possession play was very, very good and our out of possession was nullifying what's been a fantastic attacking force at Newcastle over the last 12 months or so.
"To keep that attacking force quiet, first and foremost, was the key to success."
"I think the team will grow in terms of credit if it stays where it is [in the table] and they keep winning games of football."