Evans could have won in straight sets but two double faults in the second set tie break saw the match into a decider but the Brit secured a spot in the next round courtesy of an ace after that set also went to the wire.
Elsewhere, Ward won in straight sets 7-5 6-4 against Blaz Rola.
Ward secured a break in the eleventh game of the first set before serving it out and another break of serve late in set two secured the win for the British number three.
After five months seemingly stuck in quicksand Bernard Tomic's career took a step in the right direction with a morale-boosting victory on the lush Queen's Club turf on Monday.
The 21-year-old, once tipped to become Australia's next flag bearer, had not won a main tour match since January but edged past American Tim Smyczek in the first round of the Aegon Championships - the traditional warm-up for Wimbledon.
Tomic's lean run, not helped by hip surgery after a first-round retirement against Rafa Nadal at the Australian Open, has seen his ranking slide and with points to defend over the next month he is in danger of falling out of the world's top 100.
After opening-round defeats in Rome, Madrid, Nice and at the French Open, Tomic just looked relieved to be back on a surface that he enjoys playing on.
"I'm happy because I should have lost that match," said Tomic who broke serve in the nick of time to win 6-4 3-6 7-5.
"I thought I was going to lose. Very rarely are you going to get out of a match like that when the guy's serving at 5-4 in the third set," he told reporters.
"It's funny. You can play like crap for two sets and then find three or four points and win the match."
Tomic reached the fourth round of Wimbledon last year and made the quarter-finals at the All England Club in 2011, fuelling belief that he would start to deliver on a talent that so far has not been completely fulfilled.
Controversy surrounding his coach and father John, who was given a suspended prison sentence for assaulting a practice partner last year, and run-ins with the police at home for speeding in his Ferrari, have been unwelcome distractions.
Tomic junior was also suspended from Australia's Davis Cup team in 2012 because of concerns about his attitude.
While other young guns such as Japan's Kei Nishikori, Canada's Milos Raonic and Latvia's Ernests Gulbis have begun to threaten the top order, Tomic is being left behind.
He is hopeful a good run on grass will spark him into life although his mood was hardly enthusiastic on Monday.
"I felt bad today because I just wasn't playing good and couldn't find my timing, the courts were quick," said Tomic who has been practising on grass for a week.
"It's not easy the first match on grass. It was a weird match. It was just good to get on there today and play 90 minutes and now I'll move on to my next opponent."
Defending champion Andy Murray, fresh from naming Amelie Mauresmo as his new coach, begins his campaign later in the week against the Frenchwoman's compatriot Paul-Henri Mathieu.
Switzerland's Stanislas Wawrinka is the top seed.
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