Welch became the first female referee to be appointed to officiate an EFL game and the 37-year-old from Washington, County Durham, maintained a cool head and sound judgement throughout.
"I think she was very good indeed," said Weaver, whose side's winless run in Sky Bet League Two extended to five matches.
"Important calls were made throughout and they were the right calls.
"Hopefully it's just par for the course that we see female referees and officials in the EFL. It's about time."
Both managers agreed Welch, who only took up refereeing 11 years ago, turned in a near-faultless display.
Amid typically frantic, hurly-burly fourth-tier action she was never off the pace and her only controversial decision was to turn down a first-half penalty appeal by Harrogate.
Port Vale defender Leon Legge's clattering challenge into the back of Connor Hall arguably warranted a spot-kick. Welch was perfectly placed to make her call, but opted to wave play on.
Port Vale boss Darrel Clarke agreed with his opposite number's assessment of Welch, who is in the elite category of referees and is on FIFA's international list.
She has twice refereed the Women's FA Cup final, in 2017 and 2020.
"I thought she was focused and didn't get much wrong," Clarke said after watching his side secure a fifth straight league win for the first time in over a decade.
"I was moaning about one or two decisions, but generally it was quite consistent.
"We need to see more women referees and lineswomen moving forward. They should be judged like anyone else and she's had a good start."
Amy Fearn was the first woman to referee an EFL game when she came on as an injury replacement in the 2010 Championship game between Coventry and Nottingham Forest. But Welch is the first to be given a game from the off.
David Worrall fired Port Vale ahead three minutes into the second half and Kurtis Guthrie made sure of all three points with a low drive in the 90th minute.
Worrall's opener had come against the run of play after Harrogate had dominated the first period, forcing visiting goalkeeper Scott Brown into a string of crucial saves.
Vale leapfrogged Harrogate into 15th place in the table with both sides well clear of relegation trouble.