Hamilton described Vettel's tactics as "dangerous", and the pair, albeit light-heartedly, later disagreed on whether they afforded one other enough room.
The next round of the season, the Monaco Grand Prix, takes place on May 28 in Monte Carlo.
It was a strategic race, with the decision by Mercedes to run a longer first stint, and then using the harder Pirelli compound mid-race proved inspired.
Vettel edged past the pole-sitting Hamilton at the first corner, but the British driver overtook Vettel's Ferrari with more than 20 laps left while running on a faster set of tires as Mercedes' pit strategy paid off.
"I was surprised that when I came out we were so close", Vettel said. "The auto was quick enough but the way the race happened, it wasn't meant to be. Once again fighting, hanging in there, not much missing at the end".
Victory cut the deficit to Vettel to five points at the top of the drivers' championship.
Lewis Hamilton beat Sebastian Vettel to win the Spanish Grand Prix on Sunday, tightening Formula One's intense title fight.
Equally as important, Ferrari showed that the upgrades brought by both title contenders to Spain canceled one another out. "To be able to have this battle with a four-time champion is awesome".
His excellent jump from the start allowed him to pass Hamilton at the first turn.
After Vettel and Hamilton had both pitted, Vettel hunted down Bottas now in the lead and engaged him in a lap-long duel.More news: Moors murderer Ian Brady dies aged 79
Third place victor and Red Bull driver Daniel Ricciardo of Australia, left, celebrates on the podium with second place victor and Ferrari driver Sebastian Vettel of Germany at the Spanish Formula One Grand Prix at the Barcelona Catalunya racetrack in Montmelo, Spain, Sunday, May 14, 2017.
"Every hit of that ball, just one misplaced, or one into the net can mean the whole game, so I admire their concentration and I felt I had that battle with Sebastian".
LEWIS HAMILTON lost four and a half pounds in weight in winning Sunday's Spanish GP. They brushed going through the first of two turns, with Hamilton going off momentarily as Vettel defended his position.
Vettel had kept his lead, but Hamilton waited for a straightaway to blow past him on his faster tires and never looked back. "We're very happy when we have the chance to race Mercedes".
As the German raced away from the pit lane he emerged wheel-to-wheel with Hamilton.
Hamilton was second but there was chaos behind as Bottas bumped into Kimi Raikkonen, who, in turn, ran into Max Verstappen.
Hamilton would have had to leave his challenge until late if not for some assistance from teammate Valtteri Bottas, who held up Vettel before the Finn's Mercedes broke down in a puff of smoke. Bottas nudged Raikkonen into Verstappen, with the contact damaging the front suspensions of the Ferrari and Red Bull.
"Shame for the end result, but we leave Spain knowing that we can count on a vehicle that is solid and very quick", Arrivabene said.
"What matters is that I am leading the championship after the last race of the season, and that's my thought process", Hamilton added. So I think we can be very happy, but today we're not entirely happy because the win was there, the vehicle was good enough, but the way the race happened it wasn't meant to be.
"I love tennis, and I love watching Roger Federer and Novak Djokovic in the final, and what I really admire is consistency. This is why I race and this is what got me into racing in the beginning".