A mistake by Everton keeper Jordan Pickford helped Leicester City stretch their unbeaten Premier League run to seven although they will be disappointed with a 1-1 draw on Wednesday.
Rodriguez's 30th-minute strike was a beauty but Leicester dominated for large periods, especially after the break.
Despite that it was Everton who actually finished strongly and Dominic Calvert-Lewin missed a chance to win it.
The draw left Leicester in third spot with 39 points, two points behind leaders Manchester City who have played a game fewer. Everton remain in seventh spot with 33 although they have games in hand on most of the sides above them.
Leicester again showed why the likes Pep Guardiola and Jurgen Klopp are taking them so seriously in the title race.
The statistics revealed their domination of an Everton side who themselves had the opportunity to move into the top four, above Liverpool, with a victory.
Brendan Rodgers's side had 65% possession, made almost double the amount of passes and had 18 attempts at goal compared to the eight of the hosts on a wet evening at Goodison Park.
Leicester began brightly with James Maddison whipping a long-range effort just over the crossbar.
But Rodriguez stopped Leicester in their tracks with a superb goal against the run of play.
The Colombian's attempted pass into the box was deflected and Leicester's Wesley Fofana failed to control the ball.
Rodriguez was alert and took the loose ball before sending a curling shot past Kasper Schmeichel with his right foot.
Leicester laid siege to the Everton goal after the break with their slick passing game and James Justin steered a shot wide from a good position.
They did eventually get their reward when Tielemans struck a low right-foot shot which Pickford appeared to have covered, but the ball skidded off the wet turf and bounced in off the arms of England's first-choice keeper.
It was a poor mistake by Pickford who was being watched by national manager Gareth Southgate, but he did make partial amends later with a decent save from Maddison.
Strangely Leicester's intensity dropped a little and it was Everton who began to threaten and Calvert-Lewin will be kicking himself after heading wide with an inviting target to aim for.
"We should've gone on and won the game, that was maybe two points dropped for us but they're a really good side -- but if you can't win, don't lose," Rodgers said.