Murray soon settled and got the match back on an equal footing with a break in the eighth game but he was forced into a tie-break after failing to convert two set points on the Paire serve.
In round three, theatrical Italian Fabio Fognini almost pushed the top-seeded Scot to five sets before Murray rallied from 2-5 down in the fourth set for a 6-2, 4-6, 6-1, 7-5 triumph.
He denied that the courts this year were more slippery than usual. Normally when I see my prize money, it's a little bit less than I saw on the paper.
"More slippery? I don't know if there's been more slips this year or there's just been a couple of high-profile ones", he said.
Paire hit 50 winners compared to just 25 for Murray and entertained the crowd with several attempts at diving shots, but ultimately was far too erratic.
Konta is likely to have a tougher time against 21st seed Caroline Garcia, famously once touted by Murray as a future world number one and now climbing steadily upwards.
French Open champion Jelena Ostapenko revealed on Monday: 'I think I deserve to play on a better court than Court 12.
Already a break up, a dropped service game in the third caused minor irritation for Federer, but he promptly broke back to snuff out any chance of a Dimitrov comeback.
"It feels like everyone watches Wimbledon here with Andy, it's like nothing that we have in the States, these two weeks, it's all about him", he said. "I'm going to try to play aggressively, hopefully play well, and can sneak out a win".
Former Wimbledon champion Goran Ivanisevic is another warning Murray, in his 10th consecutive Wimbledon quarter-final, not to under-estimate 24th seed Querrey, who beat South Africa's Kevin Anderson in five sets to reach the last eight.
Adding extra spice to today's drama is a growing concern over the state of the All England Club's famed Centre Court where, according to Djokovic, a hole has developed.
He and Wimbledon champion Virginia Wade were the last British man and woman to make it through to the final eight in 1973.
"Sam likes the conditions here". He obviously has a big serve and goes for his shots. "When he's standing up on the baseline, hitting forehands, dictating, he's a very risky player".
Querrey, the United States 24th seed who knocked out Novak Djokovic in 2016, made the quarter-finals for a second successive year with a 5-7, 7-6 (7/5), 6-3, 6-7 (13/11), 6-3 over unseeded Kevin Anderson of South Africa.
Rafael Nadal kept getting pushed to the brink of defeat. With Nadal out of the picture, Murray will be assured of remaining as world No. 1 with victory today.
Andy Murray is delighted to have another Briton through to the quarter-finals here and believes Johanna Konta's breakthrough can help broaden the country's interest in tennis - because some people do not like watching him play. I can look at this quarter-final in a totally relaxed fashion. Muller is enjoying a late career surge that has brought him his first two tour titles this year, but it would be a huge shock if history repeated itself.
It is only the third time in 38 years that Britain has had fourth-round representation in both the men's and women's singles.
The American was knocked out by Milos Raonic across four sets. The first, in which he defeated Jo-Wilfried Tsonga 7-5 in the decider was played over two days.