Steven Gerrard made his 600th appearance for Liverpool but was overshadowed at Anfield by Suarez, who produced a wonderful piece of skill to bring down a raking pass on his chest and round the goalkeeper in one movement to earn his team a deserved point in the 67th minute.
"His first touch from that height was an incredible piece of skill — he is a world-class striker," said Liverpool manager Brendan Rodgers.
Liverpool, however, missed a chance to climb into the top half of the standings for the first time this season, with the 18-time English champions again failing to win at home despite dominating for large periods.
Newcastle scored from one of its rare forays forward, with Yohan Cabaye volleying home superbly in the 43rd after good work by Hatem Ben Arfa, and was forced to hang on with 10 men after Fabricio Coloccini was red-carded six minutes from time.
"Suarez is a great player. He was causing us problems and if we could have wrapped him up, we would have won the game," Newcastle manager Alan Pardew said.
In the other game, Queens Park Rangers and Reading remain winless in the league after 10 games after drawing 1-1 at Loftus Road, leaving both teams in the bottom three and destined for season of struggle.
Former France striker Djibril Cisse rescued a point for the hosts by poking home a close-range finish in the 66th, cancelling out an acrobatic opener scored by Kaspars Gorkss — a former QPR player.
The point provisionally lifts QPR off the bottom of the table above Southampton, which plays on Monday against West Bromwich Albion, but the result will increase the pressure on manager Mark Hughes.
"We're working exceptionally hard on a daily basis to turn this around and we'll get there, we know that," said Hughes, who continues to receive the backing of owner Tony Fernandes. "The sooner it turns around the better, not so much in terms of staying in this league but we anticipated before the season started that we would have a really successful season."
At Anfield, Gerrard was presented with a statue of the city's red Liver Bird symbol before the match by former teammate Gary McAllister, having been compared to Superman by Rodgers this week for his regular rescue acts down the years.
However, it was Suarez who saved Liverpool this time, his goal capping a superb all-round performance that for once wasn't tainted by accusations of diving. He was also the victim of Coloccini's studs-up tackle, although replays showed the Argentina captain merely glanced the leg of Suarez.
"Most people in football know he is a threat, he plays the false No. 9 role like (Lionel) Messi does at Barcelona," Rodgers said. "He will be disappointed we did not win."
Liverpool has won just one of its six home games this season and only three in total in 2012, stopping the team making a run at the top four at present.
"I thought we were outstanding with intensity and possession," Rodgers said. "We went to sleep just before halftime for their goal, but for every other moment I thought we were brilliant. We couldn't put the chances away we had."
Newcastle could have moved sixth with a first win at Anfield since 1994, but stayed in 10th place.