The Robins triumphed 1-0 at Moor Lane with a goal from Mark Haslam two minutes before the break.
"It was a really good team effort that won the match for us but if I have to give special praise to anyone I suppose it would have to be the back four because Salford's forward line has to qualify as one of the best in non-league and that includes the Conference. They have some serious talent up front," said the Ashton co-manager.
"Some people say I made a good save towards the end but that's not true. The ball hit me. I'm too old to run around anymore."
Gareth Seddon and Sam Madeley both featured for the home side while Dale Johnson had a place on the bench.
However, Phillips maintains his side put in a far superior performance in Saturday's 1-1 draw at Hurst Cross than they did in the replay. He commented: "We played some good football at our place without getting any reward. Last night we had to grind out a victory."
Ashton now visit Evo-stik NPL first division north club Spennymoor Town on October 11 when there will be £7,500 in prize money on offer.
"Spennymoor are another team who are putting a lot of money on the table. They've got a some experienced lads but we'll have to go up there and give a good show just as we did at Salford.
"I'll have Spenny watched when they play at Northwich on Saturday and we'll learn more about them. At the end of the day, they're a division below us," said Phillips.