However, he did recognise it will be interesting to see whether Jackson's partnership with Christopher Nkunku can "flourish" when the Frenchman returns from injury.
"The unfortunate thing for Nicolas Jackson is that he follows in a long line of Chelsea strikers that have struggled," Johnson told BBC Radio 5 Live's Football Daily.
"It’s no coincidence that Chelsea last won the league when Diego Costa scored 20 league goals. Jackson does carry that burden.
"Overall he’s made a reasonable start. Don’t forget this is not one of the big marquee signings. He only cost about £32m and if it was not for a failed medical he would have joined Bournemouth in January. It is a massive step up going from a potential Bournemouth player to leading the line for Chelsea.
"You can tell he’s a raw talent. He has pace but he is definitely missing out on Nkunku. The two of them were forming something in pre-season and now that Nkunku is close to coming back to full fitness, it will be very interesting to see if their partnership can start to flourish when they are both on the pitch."
Former Chelsea assistant manager Jody Morris added: "The thing with Nicolas Jackson is that I thought he had a great pre-season. For most Chelsea fans, not only was he playing well but he was scoring goals.
"Him and Nkunku were the two standout players but the jury was still out on Jackson because we were thinking he was still young and had not played in our league.
"In the Spurs game, there were quite a few negatives, but the best thing was what it did for Jackson’s confidence as he got to score three goals.
"I'm pleased for him from a confidence perspective but he is another one I would not want to jump to conclusions too quickly about."
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