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Both variables (dependent and independent) show autocorrelation effects. Data is time-series and stationary

When I run the regression residuals appear not to be correlated. My Durbin-Watson statistic is greater than upper critical value, so there is an evidence that error terms are not positively correlated. Also when I plot ACF for errors it looks like there is no correlation there and Ljung-Box statistic is smaller than critical value.

Can I trust my regression output, are the t-statistics reliable?

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The t-statistics are reliable in the absence of autocorrelation of the errors. The fact that the residuals don't display significant autocorrelation indicates, in a not terribly rigorous way, that the autocorrelation in your dependent variable is due to the autocorrelation in your independent variable. However, it's also important to remember that the difference between statistical significance and insignificance is not itself statistically significant in many cases, e.g., a t-statistic of 1.8 vs. a t-statistic of 2.8 is a difference of 1.0, hence the lack of rigor in the statement above.

An alternative approach would be to model the data using time series analysis techniques, which, for R, are very briefly described in CRAN task view: Time Series Analysis. These techniques can get you sharper parameter estimates by explicitly modeling cross-time correlation structures, whereas, if you don't model them explicitly, you are implicitly assuming that the only such structure in the data is due to the independent variable.

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The t-statistics are un-reliable in the presence of autocorrelation of the errors. Auto-correlation in the errors can be due either insufficient lag structures in the causal variables or insufficient dependent variable lag structure. Furthermore anomalies in the error structure cause one to incorrectly accept randomness thus care should be taken to alleviate the impact of Pulses, Level Shifts, Seasonal Pulses and/or Local Time Trends that may be present but untreated. The Durbin-Watson test only reveals significant auto-correlation of lag 1 .If there is say auto-correlation of say lag S where S is the frequency of measurement (4,7,12 etc.) the DW test will incorrectly suggest randomness.

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