Watson, who came within two points of beating eventual champion Serena Williams in the third round last year, clawed her way back into the much delayed first-round clash after losing nine straight games in the second set and start of the third.
The frustration showed as Watson, 24, earned a warning for repeatedly smashing her racket at the changeover after the evenly-matched pair traded breaks in the fourth and fifth games of the third set.
Watson, world number 55, missed three match points with the score at 10-9 in the decider.
Beck, who needed treatment on her ankle after tumbling in the damp conditions before play was abandoned on Wednesday, held her nerve to break Watson and hold her own serve, reaching the second round after two hours 52 minutes on court.
Beck, ranked 43rd, will play Aliaksandra Sasnovich of Belarus in the second round.
Sasnovich beat French 31st seed Kristina Mladenovic.