Neves headed Portugal into a sixth-minute lead, rising to meet a Pedro Neto cross from the left after the Selecao das Quinas made a fast start.
DR Congo, appearing at their first World Cup since 1974 when the country competed as Zaire, grew into the contest as the half wore on.
Yoane Wissa, the Newcastle United forward, levelled in stoppage time at the end of the first half, powering home a header from an Arthur Masuaku cross to score his nation's first ever World Cup goal.
Joao Cancelo thought he had restored Portugal's lead early in the second half with an overhead kick, but the effort was ruled out for offside.
Cristiano Ronaldo, appearing in his sixth World Cup, started the match but had a peripheral influence as Portugal struggled to create clear chances despite dominating possession.
Portugal head coach Roberto Martinez explained why he persisted with his captain in a congested Congolese defence.
"We were finding it difficult because they had a back six. And in a situation like that, you can use his qualities. It makes no sense to get a player like Cristiano out of a game where you need goals."
Martinez pointed to Ronaldo's movement off the ball as a key factor in stretching the opposition.
"The straight line is not the quickest way. The way he attracts defenders and uses space is valuable. When you need goals, you need a player like Cristiano Ronaldo on the pitch."
Full-back Joao Cancelo conceded Portugal had fallen short of their attacking standards.
"We had a lot of possession, but we didn't create many clear-cut chances. We lacked intensity in the final third. We could have played better and we allowed our opponents too many transition opportunities."
Cancelo called for greater consistency as the group stage continues.
"We're all players with great humility. We need to look at where we went wrong, correct our mistakes, and win the next match."
Neves, who scored the only goal of the contest for his side, kept his focus on the collective rather than his individual milestone.
"It was my first World Cup match, and I'm happy with my performance. But, as I always say, what matters is the team. There are things we need to improve. What we really need are victories. The squad is united and determined to get better."
Wissa, who became the first player to score for DR Congo at a World Cup, spoke of his pride after the final whistle.
"I'm obviously very proud today because we've worked extremely hard. It was a very difficult match against a team that was stronger than us. But we showed courage and resilience."
The forward added that the achievement reflected the wider character of Sebastien Desabre's squad.
"Scoring our first goal is a huge source of pride because it reflects the character of this team. The most important thing now is to keep going."
Portugal sit second in Group K on a single point after the opening round of fixtures, level with DR Congo, as both sides prepare for their next group matches.