Enrique's men still need to win their two remaining games against local rivals Espanyol and Granada to be assured of winning the league after Atletico Madrid and Real Madrid were also victorious on Saturday.
Only Barca's superior head-to-head record separates them from Atletico with Real Madrid just a point behind in third.
Barca beat Betis to remain top, Atletico, Real keep up pressure
"I have the sensation that it is closer all the time," said Enrique.
"The next game we have is a derby against Espanyol and I am sure the Camp Nou will be rocking."
Barca threw away a nine-point lead over Atletico with three consecutive La Liga defeats following a 39-game unbeaten run earlier this month.
Ronaldo winner ends Barcelona's 39-game unbeaten run
However, the Catalans have responded like champions by scoring 16 goals without reply in their three league games since to keep matters in their own hands.
"This is a very difficult league with three teams in the fight to win it," added Enrique.
"I think all three of us are playing at a good level, having a great season and I don't think any of the three are going to slip up, therefore we have to do our job.
"We have the privileged situation."
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Barca were given a huge helping hand by two clear Betis errors as Heiko Westermann chopped down Ivan Rakitic to rightly see a second yellow card before half-time.
And Rakitic was left with a tap in to open the scoring five minutes after the break after a mix-up between German Pezzella and Antonio Adan.
Luis Suarez then sealed the points nine minutes from time from a sensational through ball by Lionel Messi to register his 54th goal of the season.
"I don't think we suffered at all. We were in control the whole time despite the adverse circumstances of a very dry pitch which affected both teams, Betis more than Barca," continued Enrique.
"In the move for the sending-off, if the pitch was quick the ball would have fallen to the centre-back first and I think we benefited from it in one of the goals too.
"It is always difficult to beat a well-organised team defensively like Betis.
"But in the second-half with the numerical advantage we created lots of chances and given the situation where only winning counts, I think we played well."
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