Hamilton's victory in Sunday's Spanish Grand Prix means the 29-year-old has now won the last four races, and leads the championship for the first time in almost two years.
Hamilton has taken over from Sebastian Vettel as the all-conquering force - the 26-year-old German won the last four titles and ended last year with a record run of nine successive victories.
Assessing Vettel's performances compared to those of Hamilton at present, Horner said: "With Sebastian he was just clearly a step ahead of the rest of the field. His confidence was high, he was on the crest of a wave, and then you can walk on water. He had the feeling from the car everything was going for him. Lewis probably also now feels he can walk on water. He's doing a super job.
"But if [Nico] Rosberg can get his qualifying together then he is going to give Lewis a hard time."
Fernando Alonso has already suggested Mercedes could win all 19 races this season given the performance of the car, yet Horner is slightly more sceptical.
"It's possible, but improbable," said Horner. "They have a very good engine-car-driver combination at the moment, which is all working extremely well, but things can change. We won the last nine races of last year and things changed very quickly."
It is why Horner is naturally refusing to throw in the towel on the championship, even though winning a fifth consecutive constructors' and drivers' titles appear to be beyond them.
Vettel and team-mate Daniel Ricciardo, fourth and third respectively at the Circuit de Catalunya, are currently 55 and 61 points adrift of Hamilton, while Red Bull are 113 points down on Mercedes, who have scored four successive one-twos.
"Mercedes have got themselves into a dominant position, and have worked hard to get themselves into that position," added Horner. "We have a choice: we either pack up and go home, or we fight and try to close the gap. We made small inroads in Spain and we are determined to keep pushing, keep trying to close that gap down. Obviously maximum points every weekend for them is putting them in a very strong position. But we are going to push like hell all the way through the year to the last race in Abu Dhabi."
One battle Horner potentially faces off-track this season is keeping hold of the design guru Adrian Newey, who is again being linked with a move to Ferrari. Newey has previously stated he is happy at Red Bull, in particular as his family is settled and he would be loath to move.
One final challenge may yet tempt Newey, yet Horner said: "It is an annual thing there is speculation about Adrian's future. But he has always made his feelings clear to me about the team, the paternal feeling he has towards Red Bull.
"He has been involved since the beginning, is much more involved than he has been at any other team, and he enjoys the environment he works in. The speculation is purely that, but you can understand why Ferrari would be looking."