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The top 100 cited articles of all time are featured in the October 30 issue of Nature, and statistics is one of six areas the magazine calls out. The statistics paper in this elite list include:

11. Nonparametric estimation from incomplete observations. Kaplan, E. L. & Meier, P. Journal of the American Statistical Association. 53, 457–481 (1958).

24. Regression models and life-tables. Cox, D. R. J. R. Stat. Soc., B 34, 187–220 (1972).

29. Statistical methods for assessing agreement between two methods of clinical measurement. Bland, J. M. & Altman, D. G. Lancet 327, 307–310 (1986).

57. Maximum likelihood from incomplete data via EM algorithm. Dempster, A. P., Laird, N. M. & Rubin, D. B. J. R. Stat. Soc., B 39, 1–38 (1977).

59. Controlling the false discovery rate: a practical and powerful approach to multiple testing. Benjamini, Y. & Hochberg, Y. J. R. Stat. Soc. B 57, 289–300 (1995).

64. Multiple range and multiple F tests. Duncan, D. B. Biometrics 11, 1–42 (1955).

68. The measurement of observer agreement for categorical data. Landis, J. R. & Koch, G. G. Biometrics 33, 159–174 (1977).

73. A new look at statistical-model identification. Akaike, H. IEEE Trans. Automat. Contr. 19, 716–723 (1974).

88. An algorithm for least-squares estimation of nonlinear parameters. Marquardt, D. W. J. Soc. Ind. Appl. Math. 11, 431–441 (1963).

There were also articles in the top 100 list that seemed to have considerable statistical content, especially number 41, which applied Efron's bootsrap to the accuracy of different parts of an evolutionary tree: 

41. Confidence limits on phylogenies: an approach using the bootstrap. Felsenstein, J. Evolution 39, 783–791 (1985).

Two other article titles hint at statistical content:

33. The moderator–mediator variable distinction in social psychological-research — conceptual, strategic, and statistical considerations. Baron, R. M. & Kenny, D. A. J. Pers. Soc. Psychol. 51, 1173–1182 (1986).

100. MrBayes 3: Bayesian phylogenetic inference under mixed models. Ronquist, F. & Huelsenbeck, J. P.
Bioinformatics 19, 1572–1574 (2003).

In the feature article on the top 100 list, the other five areas that are highlighted are biological techniques, bioinformatics, philogenetics, density functional theory, and crystallography.  The article is a very interesting read and includes the citation time-series for each of the articles, which suggests a dynamic list.

Quoted in the article, Stephen Stigler notes that the statistics paper in the list are not necessarily those most important to statisticians but to the wider community of scientists. The article also quotes Brad Efron for the influence of computing power on statistic theory and practice.

The prominence of articles by statisticians demonstrates the importance of statistics and statisticians to advancing science. Further showing the influence of statisticians, most-cited mathematician lists frequently feature many statisticians. For example, in this 2005 InCites list, most, if not all, of the top ten are statisticians:

Further, in a 2002 Science Watch list for most cited authors in “mathematics” for the decade 1991-2001, 20 of the 27 were statisticians.  

Also in this vein, on the most-cited PNAS articles as of November 1, 2014, two of the top ten are co-authored by statistician Rob Tibshirani:

5. Virginia Goss Tusher, Robert Tibshirani, and Gilbert Chu
Significance analysis of microarrays applied to the ionizing radiation response
PNAS 2001 98 (9) 5116-5121; published ahead of print April 17, 2001, doi:10.1073/pnas.091062498

10. John D. Storey and Robert Tibshirani
Statistical significance for genomewide studies
PNAS 2003 100 (16) 9440-9445; published ahead of print July 25, 2003, doi:10.1073/pnas.1530509100 

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