The world number one was reported to have been involved in a close shave with a truck pre-tournament but insisted wildcard Ward came closer to knocking him off on Monday.
"Yeah, actually just also recently they told me about that small article that went into papers. There is no truth to that," Djokovic said after opening his title defence with a 6-0, 7-6 (7-3), 6-4 win.
"Actually I did come few days in the beginning of the week before Wimbledon with my bicycle.
"I was asked for my own safety not to arrive with the bicycle inside of the All England Club territory.
"So from the gate of Aorangi (practise courts) where I enter with the bicycle to the transportation desk, which is a little uphill - literally 20 metres - it's very dangerous traffic around there."
Close shave or not, Djokovic was also cruising at 6-0, 3-0 against Ward before suffering a scare.
Britain's world number 177 fought back to force a second-set tiebreaker before the top seed regained control to post his 29th consecutive grand slam victory.
The reigning Wimbledon and Australian, French and US Open champion is bidding to become the first man in 78 years to win five consecutive majors.
He next faces Frenchman Adrian Mannarino on Wednesday for a place in the third round.
Big guns progress on opening day
Day one also brought wins for Rodger Federer and Milos Raonic, whiel Garbine Murguruza was forced to work for her first win of the tournament.
Federer was well short of his best, but the seven-time Wimbledon champion advanced to the second round with an uninspired 7-6 (7-5), 7-6 (7-3), 6-3 victory over Argentina's Guido Pella.
Federer has been plagued by injury problems this year and the world number three lacked rhythm and confidence in his opening match at this year's grass-court Grand Slam.
The 34-year-old, without a major title since 2012, will play British qualifier Marcus Willis for a place in the last 32.
Canadian sixth seed Raonic got off to a solid start with a 7-6 (7-4), 6-2, 6-4 win over Spain's Pablo Carreno Busta under the watchful gaze of John McEnroe.
While the big-serving 25-year-old was not at the powerful best that took him to his first grasscourt final at Queen's just over a week ago, his class overwhelmed Busta.
Much of the attention on Court Two was devoted to three-times Wimbledon champion McEnroe, who slipped into the players' box when Raonic was 3-2 up in a close first set.
Unseeded and cast out to Wimbledon's Court 16 to face American Bjorn Fratangelo, Grigor Dimitrov won 6-3, 6-4, 6-2 to reach the second round.
It snapped a sequence of five first-round defeats for the 25-year-old who is still to reach the heights many predicted after he won the Wimbledon junior title in 2008.
Murguruza grafts, Ivanovic crumbles
French Open champion Muguruza had to work hard to transfer her punch on Paris clay to Wimbledon's grass, beating Camila Giorgi 6-2, 5-7, 6-4 with a performance of powerful baseline tennis.
The Spanish world number two and her battling Italian opponent, ranked 65 places below, entertained the Centre Court with intense rallies of relentless ferocity.
It was the confidence and enormous serve of last year's runner-up Muguruza that got the better of Giorgi - but not before the 24-year-old Italian had overcome bouts of inconsistency to force a forehand error and grab the second set.
As she has done for most of the past 19 British summers, Venus Williams took a large, languid stride into the second round.
The oldest woman in the main singles draw, five-times champion Williams was at times sublime as she beat Croatian Donna Vekic 7-6 (7-3), 6-4 on a sunny Court One.
And in the tournament's first big surprise, former top-ranked Ana Ivanovic was beaten 6-2, 7-5 by Ekaterina Alexandrova, a Russian qualifier ranked 223rd in the world and making her Grand Slam debut.
The Serb blamed a wrist injury for a surprise defeat to a player who had never played in a grand slam prior to this match.